UNITED STATES
                       SECURITIES AND EXCHANGE COMMISSION
                             Washington, D.C. 20549

                                   FORM N-CSR

        CERTIFIED SHAREHOLDER REPORT OF REGISTERED MANAGEMENT INVESTMENT
                                   COMPANIES

                  Investment Company Act file number 811-22528
                                                    -----------


                     First Trust Energy Infrastructure Fund
         -------------------------------------------------------------
               (Exact name of registrant as specified in charter)


                       120 East Liberty Drive, Suite 400
                               Wheaton, IL 60187
         -------------------------------------------------------------
              (Address of principal executive offices) (Zip code)


                             W. Scott Jardine, Esq.
                          First Trust Portfolios L.P.
                       120 East Liberty Drive, Suite 400
                               Wheaton, IL 60187
         -------------------------------------------------------------
                    (Name and address of agent for service)


        registrant's telephone number, including area code: 630-765-8000
                                                           --------------

                      Date of fiscal year end: November 30
                                              -------------

                     Date of reporting period: May 31, 2014
                                              --------------


Form N-CSR is to be used by management investment companies to file reports with
the Commission not later than 10 days after the transmission to stockholders of
any report that is required to be transmitted to stockholders under Rule 30e-1
under the Investment Company Act of 1940 (17 CFR 270.30e-1). The Commission may
use the information provided on Form N-CSR in its regulatory, disclosure review,
inspection, and policymaking roles.

A registrant is required to disclose the information specified by Form N-CSR,
and the Commission will make this information public. A registrant is not
required to respond to the collection of information contained in Form N-CSR
unless the Form displays a currently valid Office of Management and Budget
("OMB") control number. Please direct comments concerning the accuracy of the
information collection burden estimate and any suggestions for reducing the
burden to Secretary, Securities and Exchange Commission, 100 F Street, NE,
Washington, DC 20549. The OMB has reviewed this collection of information under
the clearance requirements of 44 U.S.C. ss. 3507.



ITEM 1. REPORTS TO STOCKHOLDERS.

The Report to Shareholders is attached herewith.



                                   FIRST TRUST
                               -------------------
                                   E N E R G Y
                                  INFRASTRUCTURE
                               -------------------
                                      FUND
                                      (FIF)

                                                              SEMI-ANNUAL REPORT
                                                              For the Six Months
                                                              Ended May 31, 2014

                                                                     FIRST TRUST
                                                     Energy Income Partners, LLC





--------------------------------------------------------------------------------
TABLE OF CONTENTS
--------------------------------------------------------------------------------

                  FIRST TRUST ENERGY INFRASTRUCTURE FUND (FIF)
                               SEMI-ANNUAL REPORT
                                  MAY 31, 2014

Shareholder Letter..........................................................   1
At a Glance.................................................................   2
Portfolio Commentary........................................................   3
Portfolio of Investments....................................................   5
Statement of Assets and Liabilities.........................................  10
Statement of Operations.....................................................  11
Statements of Changes in Net Assets.........................................  12
Statement of Cash Flows.....................................................  13
Financial Highlights........................................................  14
Notes to Financial Statements...............................................  15
Additional Information......................................................  21


                  CAUTION REGARDING FORWARD-LOOKING STATEMENTS

This report contains certain forward-looking statements within the meaning of
the Securities Act of 1933, as amended, and the Securities Exchange Act of 1934,
as amended. Forward-looking statements include statements regarding the goals,
beliefs, plans or current expectations of First Trust Advisors L.P. ("First
Trust" or the "Advisor") and/or Energy Income Partners, LLC ("EIP" or the
"Sub-Advisor") and their respective representatives, taking into account the
information currently available to them. Forward-looking statements include all
statements that do not relate solely to current or historical fact. For example,
forward-looking statements include the use of words such as "anticipate,"
"estimate," "intend," "expect," "believe," "plan," "may," "should," "would" or
other words that convey uncertainty of future events or outcomes.

Forward-looking statements involve known and unknown risks, uncertainties and
other factors that may cause the actual results, performance or achievements of
First Trust Energy Infrastructure Fund (the "Fund") to be materially different
from any future results, performance or achievements expressed or implied by the
forward-looking statements. When evaluating the information included in this
report, you are cautioned not to place undue reliance on these forward-looking
statements, which reflect the judgment of the Advisor and/or Sub-Advisor and
their respective representatives only as of the date hereof. We undertake no
obligation to publicly revise or update these forward-looking statements to
reflect events and circumstances that arise after the date hereof.

                        PERFORMANCE AND RISK DISCLOSURE

There is no assurance that the Fund will achieve its investment objective. The
Fund is subject to market risk, which is the possibility that the market values
of securities owned by the Fund will decline and that the value of the Fund
shares may therefore be less than what you paid for them. Accordingly, you can
lose money by investing in the Fund. See "Risk Considerations" in the Additional
Information section of this report for a discussion of certain other risks of
investing in the Fund.

Performance data quoted represents past performance, which is no guarantee of
future results, and current performance may be lower or higher than the figures
shown. For the most recent month-end performance figures, please visit
http://www.ftportfolios.com or speak with your financial advisor. Investment
returns, net asset value and common share price will fluctuate and Fund shares,
when sold, may be worth more or less than their original cost.

                            HOW TO READ THIS REPORT

This report contains information that may help you evaluate your investment. It
includes details about the Fund and presents data and analysis that provide
insight into the Fund's performance and investment approach.

By reading the portfolio commentary by the portfolio management team of the
Fund, you may obtain an understanding of how the market environment affected the
Fund's performance. The statistical information that follows may help you
understand the Fund's performance compared to that of relevant market
benchmarks.

It is important to keep in mind that the opinions expressed by personnel of EIP
are just that: informed opinions. They should not be considered to be promises
or advice. The opinions, like the statistics, cover the period through the date
on the cover of this report. The risks of investing in the Fund are spelled out
in the prospectus, the statement of additional information, this report and
other Fund regulatory filings.





--------------------------------------------------------------------------------
SHAREHOLDER LETTER
--------------------------------------------------------------------------------

                  FIRST TRUST ENERGY INFRASTRUCTURE FUND (FIF)
                  SEMI-ANNUAL LETTER FROM THE CHAIRMAN AND CEO
                                  MAY 31, 2014


Dear Shareholders:

I am pleased to present you with the semi-annual report for your investment in
First Trust Energy Infrastructure Fund (the "Fund").

As a shareholder, twice a year you receive a detailed report about your
investment, including portfolio commentary from the Fund's management team, a
performance analysis and a market and Fund outlook. Additionally, First Trust
Advisors L.P. ("First Trust") compiles the Fund's financial statements for you
to review. These reports are intended to keep you up-to-date on your investment,
and I encourage you to read this document and discuss it with your financial
advisor.

As you are probably aware, the six months covered by this report saw both
challenging economic and political issues in the U.S. However, the period was
still positive for the markets. In fact, the S&P 500(R) Index, as measured on a
total return basis, rose 7.62% during the six months ended May 31, 2014. Of
course, past performance can never be an indicator of future performance, but
First Trust believes that staying invested in quality products through up and
down markets and having a long-term horizon can help investors as they work
toward their financial goals.

First Trust continues to offer a variety of products that we believe could fit
the financial plans for many investors seeking long-term investment success.
Your financial advisor can tell you about the other investments First Trust
offers that might fit your financial goals. We encourage you to discuss those
goals with your financial advisor regularly so that he or she can help keep you
on track and help you choose investments that match your goals.

First Trust will continue to make available up-to-date information about your
investments so you and your financial advisor are current on any First Trust
investments you own. We value our relationship with you, and thank you for the
opportunity to assist you in achieving your financial goals.

Sincerely,

/s/ James A. Bowen

James A. Bowen
Chairman of the Board of Trustees
Chief Executive Officer of First Trust Advisors L.P.

                                                                          Page 1





FIRST TRUST ENERGY INFRASTRUCTURE FUND (FIF)
"AT A GLANCE"
AS OF MAY 31, 2014 (UNAUDITED)

------------------------------------------------------------------
FUND STATISTICS
------------------------------------------------------------------
Symbol on New York Stock Exchange                              FIF
Common Share Price                                          $22.59
Common Share Net Asset Value ("NAV")                        $25.04
Premium (Discount) to NAV                                    (9.78)%
Net Assets Applicable to Common Shares                $439,508,941
Current Monthly Distribution per Common Share (1)          $0.1100
Current Annualized Distribution per Common Share           $1.3200
Current Distribution Rate on Closing Common Share Price (2)  5.84%
Current Distribution Rate on NAV (2)                         5.27%
------------------------------------------------------------------

------------------------------------------------------------------
         COMMON SHARE PRICE & NAV (WEEKLY CLOSING PRICE)
------------------------------------------------------------------
         Common Share Price       NAV
5/13          $23.08             $23.86
               23.50              24.02
               23.60              24.02
               21.53              23.10
6/13           23.81              24.03
               24.39              23.82
               24.10              24.98
               23.62              24.90
7/13           24.12              24.97
               25.08              24.59
               23.17              24.46
               22.25              23.79
               22.25              23.98
8/13           22.12              23.64
               22.02              23.20
               21.75              23.18
               22.00              23.75
9/13           21.68              23.72
               21.65              23.48
               21.46              23.71
               21.98              24.17
10/13          23.47              24.63
               21.27              22.56
               21.12              22.32
               21.84              22.58
               21.24              22.49
11/13          21.71              22.31
               20.31              21.92
               19.74              21.52
               19.71              22.18
12/13          20.61              22.61
               20.28              22.34
               20.16              22.41
               20.07              22.38
               20.13              22.46
1/14           20.45              22.88
               20.52              22.89
               20.87              23.36
               20.72              23.13
2/14           20.54              23.04
               20.64              23.02
               20.60              23.07
               20.84              22.98
3/14           20.97              23.41
               21.65              23.82
               21.58              23.82
               21.88              24.34
4/14           21.83              24.34
               22.25              24.53
               22.15              24.52
               22.19              24.79
               22.29              24.90
5/14           22.59              25.04




------------------------------------------------------------------------------------------------------
PERFORMANCE
------------------------------------------------------------------------------------------------------
                                                                                    Average Annual
                                                                                     Total Return
                                                                                 ---------------------
                                              6 Months Ended      1 Year Ended   Inception (9/27/2011)
                                                 5/31/2014          5/31/2014        to 5/31/2014
                                                                             
FUND PERFORMANCE (3)
NAV                                               15.82%             20.45%             22.63%
Market Value                                      7.38%              12.34%             15.99%

INEX PERFORMANCE
Philadelphia Stock Exchange Utility Index         13.12%             15.33%             11.35%
Alerian MLP Total Return Index                    11.62%             18.36%             21.10%
Blended Benchmark (4)                             12.43%             16.95%             16.39%
------------------------------------------------------------------------------------------------------


----------------------------------------------------------
                                               % OF TOTAL
INDUSTRY CLASSIFICATION                        INVESTMENTS
----------------------------------------------------------
Pipelines                                         52.0%
Electric Power                                    28.0
Natural Gas Utility                                7.4
Propane                                            5.2
Coal                                               2.9
Marine Transportation                              2.0
Gathering & Processing                             1.0
Other                                              1.5
----------------------------------------------------------
                                        Total    100.0%
                                                 ======

----------------------------------------------------------
                                               % OF TOTAL
TOP 10 HOLDINGS                                INVESTMENTS
----------------------------------------------------------
Enbridge Energy Management, LLC                    8.2%
Kinder Morgan Management, LLC                      7.2
Williams (The) Cos., Inc.                          3.9
Kinder Morgan, Inc.                                3.9
NextEra Energy, Inc.                               3.4
Southern Co.                                       3.4
UGI Corp.                                          3.2
TransCanada Corp.                                  3.1
Northeast Utilities                                3.0
Spectra Energy Corp.                               2.9
----------------------------------------------------------
                                        Total     42.2%
                                                 ======


(1)   Most recent distribution paid or declared through 5/31/2014. Subject to
      change in the future.

(2)   Distribution rates are calculated by annualizing the most recent
      distribution paid or declared through the report date and then dividing by
      Common Share price or NAV, as applicable, as of 5/31/2014. Subject to
      change in the future.

(3)   Total return is based on the combination of reinvested dividend, capital
      gain and return of capital distributions, if any, at prices obtained by
      the Dividend Reinvestment Plan and changes in NAV per share for NAV
      returns and changes in Common Share price for market value returns. Total
      returns do not reflect sales load and are not annualized for periods less
      than one year. Past performance is not indicative of future results.

(4)   The blended benchmark consists of the following: Philadelphia Stock
      Exchange Utility Index (50%) and Alerian MLP Total Return Index (50%).


Page 2





--------------------------------------------------------------------------------
PORTFOLIO COMMENTARY
--------------------------------------------------------------------------------

                  FIRST TRUST ENERGY INFRASTRUCTURE FUND (FIF)
                               SEMI-ANNUAL REPORT
                                  MAY 31, 2014

                                  SUB-ADVISOR


ENERGY INCOME PARTNERS, LLC

Energy Income Partners, LLC ("EIP" or the "Sub-Advisor"), located in Westport,
CT, was founded in 2003 to provide professional asset management services in the
area of energy-related master limited partnerships ("MLPs") and other
high-payout securities such as pipeline companies, power utilities and Canadian
income equities. EIP mainly focuses on investments in energy-related
infrastructure assets such as pipelines, power transmission and distribution,
petroleum storage and terminals that receive fee-based or regulated income from
their corporate and individual customers. EIP manages or supervises
approximately $5.5 billion of assets as of May 31, 2014. Private funds advised
by EIP include a partnership for U.S. high net worth individuals and a
master-and-feeder fund for institutions. EIP also serves as an advisor to
separately managed accounts for individuals and institutions and provides its
model portfolio to unified managed accounts and a variable insurance trust. EIP
is a registered investment advisor and serves as a Sub-Advisor to three
closed-end management investment companies in addition to the First Trust Energy
Infrastructure Fund ("FIF" or the "Fund") and an actively managed
exchange-traded fund (ETF).

                           PORTFOLIO MANAGEMENT TEAM

     JAMES J. MURCHIE                                         EVA PAO
     PORTFOLIO MANAGER                                 CO-PORTFOLIO MANAGER
    FOUNDER AND CEO OF                                     PRINCIPAL OF
ENERGY INCOME PARTNERS, LLC                         ENERGY INCOME PARTNERS, LLC


                                   COMMENTARY

FIRST TRUST ENERGY INFRASTRUCTURE FUND

The investment objective of the Fund is to seek a high level of total return
with an emphasis on current distributions paid to shareholders. The Fund pursues
its objective by investing primarily in securities of companies engaged in the
energy infrastructure sector. These companies principally include
publicly-traded master limited partnerships ("MLPs"), MLP affiliates, Canadian
income equities, pipeline companies, utilities and other infrastructure-related
companies that derive at least 50% of their revenues from operating or providing
services in support of infrastructure assets such as pipelines, power
transmission and petroleum and natural gas storage in the petroleum, natural gas
and power generation industries (collectively, "Energy Infrastructure
Companies"). Under normal market conditions, the Fund invests at least 80% of
its managed assets (total asset value of the Fund minus the sum of the Fund's
liabilities other than the principal amount of borrowing) in securities of
Energy Infrastructure Companies. There can be no assurance that the Fund's
investment objective will be achieved. The Fund may not be appropriate for all
investors.

MARKET RECAP

As measured by the Alerian MLP Total Return Index ("AMZX") and the Philadelphia
Stock Exchange Utility Index ("UTY") (together, the "MLP Benchmarks), the total
return for energy-related MLPs and utilities over the six months ended May 31,
2014, was 11.62% and 13.12%, respectively. These figures are according to data
collected from several sources, including Alerian Capital Management and
Bloomberg. While in the short term, share appreciation of Energy Infrastructure
Companies can be volatile, we believe that over the longer term, share
appreciation will approximate growth in per share quarterly cash distributions
and dividends. Over the last 10 years, growth in per share MLP distributions and
utility dividends has averaged 6.6% and 5.4%, respectively. Over the last 12
months, the cash distributions of MLPs increased by about 5.0% and utilities
decreased by about -0.8% (source: Alerian Capital Management and Bloomberg).

PERFORMANCE ANALYSIS

On a net asset value ("NAV") basis, the Fund provided a total return(1) of
15.82%, including the reinvestment of dividends, for the six months ended May
31, 2014. This compares, according to collected data, to a 12.43% return of the
Blended Benchmark. Unlike the Fund, the indices do not incur fees and expenses.
On a market value basis, the Fund had a total return, including the

(1)   Total return is based on the combination of reinvested dividend, capital
      gain and return of capital distributions, if any, at prices obtained by
      the Dividend Reinvestment Plan and changes in NAV per share for NAV
      returns and changes in Common Share price for market value returns. Total
      returns do not reflect sales load and are not annualized for periods less
      than one year. Past performance is not indicative of future results.


                                                                          Page 3




--------------------------------------------------------------------------------
PORTFOLIO COMMENTARY (CONTINUED)
--------------------------------------------------------------------------------

reinvestment of dividends, of 7.38% for the six-month reporting period. On May
31, 2014, the Fund was priced at $22.59, while the NAV was $25.04, a discount of
9.78%. On November 30, 2013, the Fund was priced at $21.71, while the NAV was
$22.30, a discount of 2.65%.

The Fund declared regular monthly Common Share distributions of $0.11 per share
for each month from December 2013 through May 2014.

The Fund's NAV outperformed the 12.43% return of the Blended Benchmark.
Contributing to this was the non-MLP portion of the portfolio which outperformed
the UTY index. Outperforming non-MLP positions included utilities, MLP parent
corporations and Canadian infrastructure corporations. The MLP portion of the
portfolio performed in-line with the MLP index. Income was enhanced by writing
covered calls on select portfolio positions.

An important factor that affected the return of the Fund was its use of
financial leverage through the use of a line of credit. The Fund has a committed
facility agreement with The Bank of Nova Scotia with a maximum commitment amount
of $165,000,000. The Fund uses leverage because the Sub-Advisor believes that,
over time, leverage can enhance total return for common shareholders. However,
the use of leverage can also increase the volatility of the NAV and therefore
volatility of the share price. For example, if the prices of securities held by
the Fund decline, the effect of changes in Common Share NAV and common
shareholder total return loss is magnified by the use of leverage. Conversely,
leverage may enhance Common Share returns during periods when the prices of
securities held by the Fund generally are rising. Unlike the Fund, AMZX and UTY
are not leveraged. Leverage had a positive impact on the performance of the Fund
over this reporting period.

MARKET AND FUND OUTLOOK

MLPs continue to play an integral role in the restructuring of more diversified
energy conglomerates. This restructuring includes the creation by these more
diversified conglomerates of subsidiaries with high dividend payout ratios that
contain assets with stable cash flows such as pipelines, storage terminals and
electric power assets with long-term fixed-price contracts. The restructuring
can also include the divestiture by some of the parent companies of most or all
of their cyclical businesses, leaving the parent company looking very similar to
an old-fashioned pipeline utility with a large holding in a subsidiary MLP. In
our view, these diversified energy conglomerates are restructuring so that their
regulated infrastructure assets that have predictable cash flows may be better
valued by the market. This results in a better financing tool to raise capital
for the new energy infrastructure projects related to the rapid growth of North
American oil and gas production. This phenomenon is beginning to spread to the
power utility industry but instead of spinning out an MLP, diversified power
companies are spinning out a regular "C" corporation with a higher dividend
payout ratio (relative to earnings) and a resulting higher yield. Three such
"Yield-Co.'s" have been created in the last year and EIP expects more will be
created.

The MLP asset class has experienced four Initial Public Offerings ("IPOs") in
2014, as of May 31, 2014. In addition, there was secondary financing activity
for MLPs during the reporting period as MLPs continued to fund their ongoing
investments in new pipelines, processing and storage facilities. There have been
23 secondary equity offerings for MLPs, as of May 31, 2014, which raised $7.2
billion in proceeds. This compares to $20.3 billion raised in all of 2013. MLPs
also found access to the public debt markets, raising $13.4 billion in 15
offerings through May 31, 2014. This compares to $19.9 billion in calendar year
2013 (source: Barclays Capital).

Capital spending for utilities has remained relatively unchanged. As measured by
UTY, capital expenditures for the power utility industry were $81.8 billion in
2012 and $80.8 billion in 2013. The expenditures are in response to needs such
as reliability, interconnection, modernization and growing demand. These capital
investments are supported, in part, by federal and state regulation, which
allows companies to recoup investments they have made in the rates they charge
their customers.

The Fund continues to weight the portfolio toward Energy Infrastructure
Companies with mostly non-cyclical cash flows, investment-grade ratings,
conservative balance sheets, modest and/or flexible organic growth commitments
and liquidity on their revolving lines of credit. Since the Fund invests in
securities that tend to have high dividend payout ratios (as measured versus
earnings), securities with unpredictable cyclical cash flows make them a poorer
fit with the portfolio, in the opinion of the Sub-Advisor. While there are some
businesses within the Fund's portfolio whose cash flows are cyclical, they are
usually small and analyzed in the context of each company's financial and
operating leverage and payout ratio.


Page 4





FIRST TRUST ENERGY INFRASTRUCTURE FUND (FIF)
PORTFOLIO OF INVESTMENTS
MAY 31, 2014 (UNAUDITED)



   SHARES                                              DESCRIPTION                                              VALUE
-------------   -----------------------------------------------------------------------------------------   --------------
COMMON STOCKS - 97.2%
                                                                                                      
                ELECTRIC UTILITIES - 22.6%
      163,400   American Electric Power Co., Inc. (a)....................................................   $    8,717,390
       78,200   Duke Energy Corp. (a)....................................................................        5,558,456
      160,400   Emera, Inc. (CAD) (a)....................................................................        5,034,042
      176,800   Fortis, Inc. (CAD) (a)...................................................................        5,305,794
       62,000   IDACORP, Inc.............................................................................        3,399,460
      255,800   ITC Holdings Corp. (a)...................................................................        9,362,280
      200,300   NextEra Energy, Inc. (a).................................................................       19,501,208
      376,600   Northeast Utilities (a)..................................................................       17,097,640
      132,400   NRG Yield, Inc., Class A (a).............................................................        6,210,884
      434,900   Southern (The) Co. (a)...................................................................       19,039,922
                                                                                                            --------------
                                                                                                                99,227,076
                                                                                                            --------------
                GAS UTILITIES - 8.9%
       50,800   Atmos Energy Corp. (a)...................................................................        2,545,080
      119,385   Laclede Group, Inc. (a)..................................................................        5,572,892
      118,000   One Gas, Inc.............................................................................        4,322,340
      359,500   Questar Corp. (a)........................................................................        8,653,165
      367,479   UGI Corp. (a)............................................................................       17,885,203
                                                                                                            --------------
                                                                                                                38,978,680
                                                                                                            --------------
                MULTI-UTILITIES - 18.2%
       49,000   Alliant Energy Corp......................................................................        2,856,700
      104,200   ATCO, Ltd., Class I (CAD) (a)............................................................        4,971,194
      166,000   Canadian Utilities, Ltd., Class A (CAD) (a)..............................................        6,008,946
      286,000   CMS Energy Corp. (a).....................................................................        8,508,500
      221,900   Dominion Resources, Inc. (a).............................................................       15,302,224
      160,500   National Grid PLC, ADR (a)...............................................................       11,994,165
      145,000   NiSource, Inc............................................................................        5,418,650
       72,000   Public Service Enterprise Group, Inc.....................................................        2,805,120
      108,000   Scana Corp...............................................................................        5,616,000
       59,000   Sempra Energy (a)........................................................................        5,920,650
      235,100   Wisconsin Energy Corp. (a)...............................................................       10,701,752
                                                                                                            --------------
                                                                                                                80,103,901
                                                                                                            --------------
                OIL, GAS & CONSUMABLE FUELS - 47.5%
    1,564,074   Enbridge Energy Management, LLC (a) (b)..................................................       46,656,327
      458,600   Enbridge Income Fund Holdings, Inc. (CAD) (a)............................................       11,631,006
      240,569   Enbridge, Inc. (a).......................................................................       11,424,622
      417,000   Inter Pipeline, Ltd. (CAD) (a)...........................................................       12,218,104
       40,300   Keyera Corp. (CAD) (a)...................................................................        2,729,161
      656,500   Kinder Morgan, Inc. (a)..................................................................       21,920,535
      568,084   Kinder Morgan Management, LLC (a) (b)....................................................       40,941,814
      133,100   Pembina Pipeline Corp. (CAD) (a).........................................................        5,255,013
      399,400   Spectra Energy Corp. (a).................................................................       16,207,652
      374,500   TransCanada Corp. (a)....................................................................       17,470,425
      476,700   Williams (The) Cos., Inc. (a)............................................................       22,385,832
                                                                                                            --------------
                                                                                                               208,840,491
                                                                                                            --------------
                TOTAL COMMON STOCKS......................................................................      427,150,148
                (Cost $377,507,240)                                                                         --------------



                        See Notes to Financial Statements                 Page 5





FIRST TRUST ENERGY INFRASTRUCTURE FUND (FIF)
PORTFOLIO OF INVESTMENTS (CONTINUED)
MAY 31, 2014 (UNAUDITED)



   SHARES/
    UNITS                                              DESCRIPTION                                              VALUE
-------------   -----------------------------------------------------------------------------------------   --------------
MASTER LIMITED PARTNERSHIPS - 31.9%
                                                                                                      
                GAS UTILITIES - 2.7%
      189,764   AmeriGas Partners, L.P. (a)..............................................................   $    8,991,018
       60,000   Suburban Propane Partners, L.P. (a)......................................................        2,785,200
                                                                                                            --------------
                                                                                                                11,776,218
                                                                                                            --------------

                OIL, GAS & CONSUMABLE FUELS - 29.2%
       24,000   Access Midstream Partners, L.P. (a)......................................................        1,511,760
       44,000   Alliance Holdings GP, L.P. (a)...........................................................        2,829,200
      132,715   Alliance Resource Partners, L.P. (a).....................................................       12,070,429
      255,200   El Paso Pipeline Partners, L.P. (a)......................................................        8,738,048
      117,000   Energy Transfer Equity, L.P. (a).........................................................        5,962,320
      151,400   Energy Transfer Partners, L.P. (a).......................................................        8,526,848
      114,600   Enterprise Products Partners, L.P. (a)...................................................        8,574,372
       73,700   EQT Midstream Partners, L.P. (a).........................................................        6,053,718
      180,776   Holly Energy Partners, L.P. (a)..........................................................        6,388,624
       75,600   Magellan Midstream Partners, L.P. (a)....................................................        6,190,128
      120,239   Natural Resource Partners, L.P. (a)......................................................        1,703,787
      103,472   NGL Energy Partners, L.P. (a)............................................................        4,138,880
      164,200   ONEOK Partners, L.P. (a).................................................................        9,047,420
        8,100   Phillips 66 Partners, L.P. (a)...........................................................          490,455
      126,686   Plains All American Pipeline, L.P........................................................        7,153,959
      156,500   Spectra Energy Partners, L.P. (a)........................................................        8,208,425
      162,095   TC Pipelines, L.P. (a)...................................................................        8,428,940
      249,628   Teekay LNG Partners, L.P. (a)............................................................       11,058,520
      172,762   TransMontaigne Partners, L.P. (a)........................................................        8,574,178
       50,000   Williams Partners, L.P. (a)..............................................................        2,655,500
                                                                                                            --------------
                                                                                                               128,305,511
                                                                                                            --------------
                TOTAL MASTER LIMITED PARTNERSHIPS........................................................      140,081,729
                (Cost $99,816,660)                                                                          --------------

                TOTAL INVESTMENTS - 129.1%...............................................................      567,231,877
                (Cost $477,323,900) (c)                                                                     --------------





  NUMBER OF
  CONTRACTS                                            DESCRIPTION                                              VALUE
-------------   -----------------------------------------------------------------------------------------   --------------
CALL OPTIONS WRITTEN - (0.7%)
                                                                                                      
                American Electric Power Co., Inc. Calls
          200   @ $55.00 due June 2014...................................................................           (2,000)
          100   @  55.00 due November 2014...............................................................          (10,000)
          600   @  60.00 due January 2015................................................................          (16,500)
                                                                                                            --------------
                                                                                                                   (28,500)
                                                                                                            --------------
                CMS Energy Corp. Calls
          280   @  30.00 due June 2014...................................................................           (5,600)
        1,400   @  32.50 due September 2014..............................................................          (17,021)
                                                                                                            --------------
                                                                                                                   (22,621)
                                                                                                            --------------
                Dominion Resources, Inc. Call
        1,330   @  75.00 due July 2014...................................................................           (9,310)
                                                                                                            --------------

Page 6                  See Notes to Financial Statements





FIRST TRUST ENERGY INFRASTRUCTURE FUND (FIF)
PORTFOLIO OF INVESTMENTS (CONTINUED)
MAY 31, 2014 (UNAUDITED)

  NUMBER OF
  CONTRACTS                                            DESCRIPTION                                              VALUE
-------------   -----------------------------------------------------------------------------------------   --------------
CALL OPTIONS WRITTEN (CONTINUED)

                Duke Energy Corp. Call
          280   @ $77.50 due July 2014...................................................................   $       (2,800)
                                                                                                            --------------
                Enbridge, Inc. Calls
        1,520   @  45.00 due July 2014...................................................................         (440,800)
          106   @  47.50 due July 2014...................................................................          (11,577)
                                                                                                            --------------
                                                                                                                  (452,377)
                                                                                                            --------------
                Kinder Morgan, Inc. Calls
          860   @  40.00 due June 2014...................................................................           (2,580)
        2,000   @  35.00 due September 2014..............................................................         (108,000)
          800   @  40.00 due September 2014..............................................................           (8,000)
          200   @  37.50 due December 2014...............................................................           (7,000)
                                                                                                            --------------
                                                                                                                  (125,580)
                                                                                                            --------------
                National Grid PLC, ADR Calls
          415   @  70.00 due June 2014...................................................................         (184,675)
          630   @  70.00 due September 2014..............................................................         (267,750)
                                                                                                            --------------
                                                                                                                  (452,425)
                                                                                                            --------------
                NextEra Energy, Inc. Call
        1,480   @  100.00 due June 2014..................................................................          (66,600)
                                                                                                            --------------
                NiSource, Inc. Calls
        1,300   @  36.00 due July 2014...................................................................         (224,250)
          150   @  37.00 due July 2014...................................................................          (16,125)
                                                                                                            --------------
                                                                                                                  (240,375)
                                                                                                            --------------
                Northeast Utilities Call
        2,260   @  50.00 due October 2014...............................................................          (56,500)
                                                                                                            --------------
                Plains All American Pipeline, L.P. Calls
          370   @  57.50 due June 2014..................................................................           (9,250)
          380   @  60.00 due November 2014..............................................................          (26,600)
                                                                                                            --------------
                                                                                                                   (35,850)
                                                                                                            --------------
                Public Service Enterprise Group, Inc. Call
          400   @  45.00 due December 2014...............................................................           (8,000)
                                                                                                            --------------
                Questar Corp. Call
        2,250   @  25.00 due July 2014...................................................................          (33,750)
                                                                                                            --------------
                Scana Corp. Calls
          200   @  55.00 due August 2014.................................................................           (3,500)
          350   @  55.00 due November 2014...............................................................          (18,375)
                                                                                                            --------------
                                                                                                                   (21,875)
                                                                                                            --------------
                Sempra Energy Call
          300   @  105.00 due July 2014..................................................................          (12,000)
                                                                                                            --------------
                Southern (The) Co. Calls
        1,120   @  46.00 due June 2014...................................................................           (5,600)
        1,000   @  45.00 due August 2014.................................................................          (36,000)
                                                                                                            --------------
                                                                                                                   (41,600)
                                                                                                            --------------

                        See Notes to Financial Statements                 Page 7





FIRST TRUST ENERGY INFRASTRUCTURE FUND (FIF)
PORTFOLIO OF INVESTMENTS (CONTINUED)
MAY 31, 2014 (UNAUDITED)


  NUMBER OF
  CONTRACTS                                            DESCRIPTION                                              VALUE
-------------   -----------------------------------------------------------------------------------------   --------------
CALL OPTIONS WRITTEN (CONTINUED)

                Spectra Energy Corp. Calls
        2,080   @ $38.00 due June 2014...................................................................   $     (556,400)
          100   @  39.00 due June 2014...................................................................          (17,250)
          210   @  39.00 due September 2014..............................................................          (46,725)
                                                                                                            --------------
                                                                                                                  (620,375)
                                                                                                            --------------
                TransCanada Corp. Calls
          170   @  47.50 due June 2014...................................................................           (2,102)
        1,510   @  50.00 due August 2014.................................................................          (18,875)
          560   @  50.00 due November 2014...............................................................          (29,400)
                                                                                                            --------------
                                                                                                                   (50,377)
                                                                                                            --------------
                UGI Corp. Call
        1,550   @  50.00 due October 2014................................................................         (139,500)
                                                                                                            --------------
                Williams (The) Cos., Inc. Calls
        1,000   @  47.00 due July 2014...................................................................          (98,000)
        1,860   @  46.00 due August 2014.................................................................         (359,910)
                                                                                                            --------------
                                                                                                                  (457,910)
                                                                                                            --------------
                Wisconsin Energy Corp. Calls
        1,090   @  45.00 due July 2014...................................................................         (129,710)
          300   @  50.00 due July 2014...................................................................           (3,750)
                                                                                                            --------------
                                                                                                                  (133,460)
                                                                                                            --------------
                TOTAL CALL OPTIONS WRITTEN...............................................................       (3,011,785)
                (Premiums received $1,647,782)                                                              --------------

                OUTSTANDING LOAN - (33.2%)...............................................................     (145,900,000)
                NET OTHER ASSETS AND LIABILITIES - 4.8%..................................................       21,188,849
                                                                                                            --------------
                NET ASSETS - 100.0%......................................................................   $  439,508,941
                                                                                                            ==============


-----------------------------

(a)   All or a portion of this security serves as collateral on the outstanding
      loan.

(b)   Non-income producing security which pays in-kind distributions. For the
      six months ended May 31, 2014, the Fund received 57,482 shares of Enbridge
      Energy Management, LLC and 20,202 shares of Kinder Morgan Management, LLC.

(c)   Aggregate cost for financial reporting purposes, which approximates the
      aggregate cost for federal income tax purposes. As of May 31, 2014, the
      aggregate gross unrealized appreciation for all securities in which there
      was an excess of value over tax cost was $94,974,614 and the aggregate
      gross unrealized depreciation for all securities in which there was an
      excess of tax cost over value was $5,066,637.

ADR   American Depositary Receipt

CAD   Canadian Dollar - Security is denominated in Canadian Dollars and is
      translated into U.S. Dollars based upon the current exchange rate.




INTEREST RATE SWAP AGREEMENTS:

                                                              NOTIONAL
   COUNTERPARTY       FLOATING RATE (1)   EXPIRATION DATE      AMOUNT      FIXED RATE (1)       VALUE
---------------------------------------------------------------------------------------------------------
                                                                             
Bank of Nova Scotia     1 month LIBOR          10/08/20     $ 36,475,000       2.121%       $    (685,474)



(1)   The Fund pays the fixed rate and receives the floating rate. The floating
      rate on May 31, 2014 was 0.151%.


Page 8                  See Notes to Financial Statements





FIRST TRUST ENERGY INFRASTRUCTURE FUND (FIF)
PORTFOLIO OF INVESTMENTS (CONTINUED)
MAY 31, 2014 (UNAUDITED)

-----------------------------

VALUATION INPUTS

A summary of the inputs used to value the Fund's investments as of May 31, 2014
is as follows (see Note 2A - Portfolio Valuation in the Notes to Financial
Statements):



                                                        ASSETS TABLE
                                                                                               LEVEL 2           LEVEL 3
                                                            TOTAL            LEVEL 1         SIGNIFICANT       SIGNIFICANT
                                                           VALUE AT           QUOTED          OBSERVABLE       UNOBSERVABLE
INVESTMENTS                                               5/31/2014           PRICES            INPUTS            INPUTS
--------------------------------------------------      --------------    --------------    --------------    --------------
                                                                                                  
Common Stocks*....................................      $  427,150,148    $  427,150,148    $           --    $           --
Master Limited Partnerships *.....................         140,081,729       140,081,729                --                --
                                                        --------------    --------------    --------------    --------------
Total.............................................      $  567,231,877    $  567,231,877    $           --    $           --
                                                        ==============    ==============    ==============    ==============

                                                      LIABILITIES TABLE
                                                                                               LEVEL 2           LEVEL 3
                                                            TOTAL            LEVEL 1         SIGNIFICANT       SIGNIFICANT
                                                           VALUE AT           QUOTED          OBSERVABLE       UNOBSERVABLE
                                                          5/31/2014           PRICES            INPUTS            INPUTS
                                                        --------------    --------------    --------------    --------------
Call Options Written..............................      $   (3,011,785)   $   (3,011,785)   $           --    $           --
Interest Rate Swap** .............................            (685,474)               --          (685,474)               --
                                                        --------------    --------------    --------------    --------------
Total ............................................      $   (3,697,259)   $   (3,011,785)   $     (685,474)   $           --
                                                        ==============    ==============    ==============    ==============


*   See Portfolio of Investments for industry breakout.
** See Interest Rate Swap Agreements for contract detail.

All transfers in and out of the Levels during the period are assumed to be
transferred on the last day of the period at their current value. There were no
transfers between Levels at May 31, 2014.


                  See Notes to Financial Statements                       Page 9





FIRST TRUST ENERGY INFRASTRUCTURE FUND (FIF)
STATEMENT OF ASSETS AND LIABILITIES
MAY 31, 2014 (UNAUDITED)



ASSETS:
                                                                                                
Investments, at value
    (Cost $477,323,900)                                                                            $ 567,231,877
Cash                                                                                                  19,704,306
Cash segregated as collateral for open swap contracts                                                  1,443,655
Receivables:
    Dividends.................................................................................         1,316,351
    Investment securities sold................................................................           104,953
    Interest..................................................................................               199
Prepaid expenses                                                                                          21,048
                                                                                                   -------------
    Total Assets..............................................................................       589,822,389
                                                                                                   -------------
LIABILITIES:
Outstanding loan                                                                                     145,900,000
Options written, at value (Premiums received $1,647,782)                                               3,011,785
Swap contracts, at value (Cost $569)                                                                     685,474
Payables:
    Investment advisory fees..................................................................           492,905
    Interest and fees on loan.................................................................            97,564
    Administrative fees.......................................................................            40,503
    Audit and tax fees........................................................................            29,751
    Custodian fees............................................................................            15,727
    Legal fees................................................................................            13,178
    Printing fees.............................................................................            11,828
    Trustees' fees and expenses...............................................................             6,346
    Transfer agent fees.......................................................................             6,030
    Financial reporting fees..................................................................               771
Other liabilities                                                                                          1,586
                                                                                                   -------------
    Total Liabilities.........................................................................       150,313,448

NET ASSETS                                                                                         $ 439,508,941
                                                                                                   =============

NET ASSETS CONSIST OF:
Paid-in capital...............................................................................     $ 333,654,042
Par value.....................................................................................           175,502
Accumulated net investment income (loss)......................................................        (3,547,526)
Accumulated net realized gain (loss) on investments, written options, swap
 contracts and foreign currency transactions..................................................        21,368,092
Net unrealized appreciation (depreciation) on investments, written options, swap
 contracts and foreign currency translation...................................................        87,858,831

NET ASSETS                                                                                         $ 439,508,941
                                                                                                   =============
NET ASSET VALUE, per Common Share (par value $0.01 per Common Share)                               $       25.04
                                                                                                   =============
Number of Common Shares outstanding (unlimited number of Common Shares has been authorized)...        17,550,236
                                                                                                   =============




Page 10                        See Notes to Financial Statements






FIRST TRUST ENERGY INFRASTRUCTURE FUND (FIF)
STATEMENT OF OPERATIONS
FOR THE SIX MONTHS ENDED MAY 31, 2014 (UNAUDITED)



INVESTMENT INCOME:
                                                                                                
Dividends (net of foreign withholding tax of $234,109)........................................     $   5,853,922
Interest......................................................................................               551
                                                                                                   -------------
    Total investment income...................................................................         5,854,473
                                                                                                   -------------
EXPENSES:
Investment advisory fees......................................................................         2,758,658
Interest and fees on loan.....................................................................           578,179
Administrative fees...........................................................................           225,866
Printing fees.................................................................................            50,973
Custodian fees................................................................................            27,655
Audit and tax fees............................................................................            25,292
Transfer agent fees...........................................................................            15,482
Legal fees....................................................................................            13,821
Trustees' fees and expenses...................................................................            12,478
Financial reporting fees......................................................................             4,625
Other.........................................................................................            25,712
                                                                                                   -------------
    Total expenses............................................................................         3,738,741
                                                                                                   -------------
NET INVESTMENT INCOME (LOSS)..................................................................         2,115,732
                                                                                                   -------------
NET REALIZED AND UNREALIZED GAIN (LOSS):
Net realized gain (loss) on:
    Investments...............................................................................        22,563,537
    Written options...........................................................................          (642,929)
    Swap contracts............................................................................          (359,465)
    Foreign currency transactions.............................................................            56,996
                                                                                                   -------------
Net realized gain (loss)......................................................................        21,618,139
                                                                                                   -------------
Net change in unrealized appreciation (depreciation) on:
    Investments...............................................................................        38,106,560
    Written options ..........................................................................        (1,683,462)
    Swap contracts............................................................................          (404,619)
    Foreign currency translation..............................................................             3,357
                                                                                                   -------------
Net change in unrealized appreciation (depreciation)..........................................        36,021,836
                                                                                                   -------------
NET REALIZED AND UNREALIZED GAIN (LOSS).......................................................        57,639,975
                                                                                                   -------------
NET INCREASE (DECREASE)  IN NET ASSETS RESULTING FROM OPERATIONS..............................     $  59,755,707
                                                                                                   =============



                 See Notes to Financial Statements                       Page 11





FIRST TRUST ENERGY INFRASTRUCTURE FUND (FIF)
STATEMENTS OF CHANGES IN NET ASSETS



                                                                                          SIX MONTHS
                                                                                            ENDED               YEAR
                                                                                          5/31/2014            ENDED
                                                                                         (UNAUDITED)         11/30/2013
                                                                                        --------------     --------------
OPERATIONS:
                                                                                                     
Net investment income (loss).......................................................     $    2,115,732     $    3,947,462
Net realized gain (loss)...........................................................         21,618,139         49,255,462
Net increase from payment by the Sub-Advisor (a)...................................                 --              5,421
Net change in unrealized appreciation (depreciation)...............................         36,021,836         11,545,110
                                                                                        --------------     --------------
Net increase (decrease) in net assets resulting from operations....................         59,755,707         64,753,455
                                                                                        --------------     --------------
DISTRIBUTIONS TO SHAREHOLDERS FROM:
Net investment income..............................................................        (11,583,156)       (14,525,290)
Net realized gain..................................................................                 --        (57,064,592)
Return of capital..................................................................                 --           (918,918)
                                                                                        --------------     --------------
Total distributions to shareholders................................................        (11,583,156)       (72,508,800)
                                                                                        --------------     --------------
Total increase (decrease) in net assets............................................         48,172,551         (7,755,345)

NET ASSETS:
Beginning of period................................................................        391,336,390        399,091,735
                                                                                        --------------     --------------
End of period......................................................................     $  439,508,941     $  391,336,390
                                                                                        ==============     ==============
Accumulated net investment income (loss) at end of period..........................     $   (3,547,526)    $    5,919,898
                                                                                        ==============     ==============
COMMON SHARES:
Common Shares at end of period.....................................................         17,550,236         17,550,236
                                                                                        ==============     ==============


------------------------------

(a) See Note 3 in the Notes to Financial Statements.


Page 12                    See Notes to Financial Statements




FIRST TRUST ENERGY INFRASTRUCTURE FUND (FIF)
STATEMENT OF CASH FLOWS
FOR THE SIX MONTHS ENDED MAY 31, 2014 (UNAUDITED)



CASH FLOWS FROM OPERATING ACTIVITIES:
                                                                                                   
Net increase (decrease) in net assets resulting from operations....................     $   59,755,707
Adjustments to reconcile net increase (decrease) in net assets resulting
  from operations to net cash provided by operating activities:

    Purchases of investments.......................................................        (94,227,699)
    Sales of investments...........................................................        109,433,650
    Proceeds from written options..................................................          2,642,536
    Amount paid to close written options...........................................         (2,460,081)
    Return of capital received from investment in MLPs.............................          3,772,153
    Net realized gain/loss on investments and written options......................        (21,920,608)
    Net change in unrealized appreciation/depreciation on investments and
      written options..............................................................        (36,423,098)
    Net change in unrealized appreciation/depreciation on swap contracts                       404,619

CHANGES IN ASSETS AND LIABILITIES:
    Increase in cash segregated as collateral for open swap contracts..............           (255,399)
    Increase in interest receivable................................................               (199)
    Decrease in dividends receivable (a)...........................................             51,814
    Increase in prepaid expenses...................................................             (8,594)
    Increase in interest and fees on loan payable..................................              1,828
    Increase in investment advisory fees payable...................................             47,224
    Decrease in audit and tax fees payable.........................................            (20,549)
    Decrease in legal fees payable.................................................             (2,929)
    Decrease in printing fees payable..............................................            (14,749)
    Increase in administrative fees payable........................................              2,727
    Decrease in custodian fees payable.............................................            (24,455)
    Increase in transfer agent fees payable........................................              3,169
    Increase in Trustees' fees and expenses payable................................                456
    Increase in other liabilities payable..........................................                439
                                                                                        --------------
CASH PROVIDED BY OPERATING ACTIVITIES..............................................                      $   20,757,962
                                                                                                         --------------

CASH FLOWS FROM FINANCING ACTIVITIES:
    Distributions to Common Shareholders from net investment income................        (11,583,156)
                                                                                        --------------
CASH USED BY FINANCING ACTIVITIES..................................................                         (11,583,156)
                                                                                                         --------------
Increase in cash...................................................................                           9,174,806
Cash at beginning of period........................................................                          10,529,500
                                                                                                         --------------
Cash at end of period..............................................................                      $   19,704,306
                                                                                                         ==============
SUPPLEMENTAL DISCLOSURE OF CASH FLOW INFORMATION:
Cash paid during the period for interest and fees..................................                      $      576,351
                                                                                                         ==============


-----------------------------

(a)   Includes net change in unrealized appreciation (depreciation) on foreign
      currency of $3,357.


                 See Notes to Financial Statements                       Page 13





FIRST TRUST ENERGY INFRASTRUCTURE FUND (FIF)
FINANCIAL HIGHLIGHTS
FOR A COMMON SHARE OUTSTANDING THROUGHOUT EACH PERIOD



                                                             SIX MONTHS
                                                                ENDED           YEAR            YEAR             PERIOD
                                                              5/31/2014         ENDED           ENDED            ENDED
                                                             (UNAUDITED)     11/30/2013      11/30/2012      11/30/2011 (a)
                                                            -------------   -------------   -------------   ----------------
                                                                                                
Net asset value, beginning of period........................  $   22.30       $   22.74       $   21.38       $   19.10 (b)
                                                              ---------       ---------       ---------       ---------
INCOME FROM INVESTMENT OPERATIONS:
Net investment income (loss)................................       0.12            0.22            0.27            0.05
Net realized and unrealized gain (loss).....................       3.28            3.47            2.38            2.30
                                                              ---------       ---------       ---------       ---------
Total from investment operations............................       3.40            3.69            2.65            2.35
                                                              ---------       ---------       ---------       ---------
Common Shares offering costs charged to paid-in
    capital.................................................         --              --            0.01           (0.04)
                                                              ---------       ---------       ---------       ---------
Capital reduction from issuance of Common Shares
    related to over allotment...............................         --              --              --           (0.03)
                                                              ---------       ---------       ---------       ---------
Distributions paid to shareholders from:
Net investment income.......................................      (0.66)          (0.83)          (1.01)             --
Net realized gain...........................................         --           (3.25)          (0.29)             --
Return of capital...........................................         --           (0.05)             --              --
                                                              ---------       ---------       ---------       ---------
Total distributions to Common Shareholders..................      (0.66)          (4.13)          (1.30)             --
                                                              ---------       ---------       ---------       ---------
Net asset value, end of period.............................   $   25.04       $   22.30       $   22.74       $   21.38
                                                              =========       =========       =========       =========
Market value, end of period................................   $   22.59       $   21.71       $   21.34       $   19.82
                                                              =========       =========       =========       =========
TOTAL RETURN BASED ON NET ASSET VALUE (c)...................      15.82%          17.76% (d)      13.08% (d)      11.94%
                                                              =========       =========       =========       =========
TOTAL RETURN BASED ON MARKET VALUE (c)......................       7.38%          22.11%          14.47%          (0.90)%
                                                              =========       =========       =========       =========
--------------------

RATIOS TO AVERAGE NET ASSETS/SUPPLEMENTAL DATA:
Net assets, end of period (in 000's)........................  $ 439,509       $ 391,336       $ 399,092       $ 375,232
Ratio of total expenses to average net assets...............       1.84% (e)       1.84%           1.82%           1.66% (e)
Ratio of total expenses to average net assets excluding
    interest expense and fees on loan.......................       1.56% (e)       1.55%           1.52%           1.48% (e)
Ratio of net investment income (loss) to average net
    assets..................................................       1.04% (e)       0.95%           1.21%           1.49% (e)
Portfolio turnover rate.....................................         17%             54%             46%              8%

INDEBTEDNESS:
Total loan outstanding (in 000's)...........................  $ 145,900       $ 145,900       $ 141,900       $ 102,000
Asset coverage per $1,000 of indebtedness (f)...............  $   4,012       $   3,682       $   3,812       $   4,679

--------------------


(a)   Initial seed date of August 18, 2011. The Fund commenced operations on
      September 27, 2011.

(b)   Net of sales load of $0.90 per Common Share on initial offering.

(c)   Total return is based on the combination of reinvested dividend, capital
      gain and return of capital distributions, if any, at prices obtained by
      the Dividend Reinvestment Plan, and changes in net asset value per share
      for net asset value returns and changes in Common Share price for market
      value returns. Total returns do not reflect sales load and are not
      annualized for periods less than one year. Past performance is not
      indicative of future results.

(d)   During the years ended November 30, 2013 and 2012, the Fund received
      reimbursements from the Sub-Advisor in the amounts of $5,421 and $104,
      respectively, in connection with trade errors. The reimbursements from the
      Sub-Advisor represent less than $0.01 per share and had no effect on the
      Fund's total returns.

(e)   Annualized.

(f)   Calculated by taking the Fund's total assets less the Fund's total
      liabilities (not including the loan outstanding) and dividing by the
      outstanding loan balance in 000's.


Page 14
                        See Notes to Financial Statements





--------------------------------------------------------------------------------
NOTES TO FINANCIAL STATEMENTS
--------------------------------------------------------------------------------

                  FIRST TRUST ENERGY INFRASTRUCTURE FUND (FIF)
                            MAY 31, 2014 (UNAUDITED)


                                1. ORGANIZATION

First Trust Energy Infrastructure Fund (the "Fund") is a non-diversified,
closed-end management investment company organized as a Massachusetts business
trust on February 22, 2011 and is registered with the Securities and Exchange
Commission under the Investment Company Act of 1940, as amended (the "1940
Act"). The Fund trades under the ticker symbol FIF on the New York Stock
Exchange ("NYSE").

The Fund's investment objective is to seek a high level of total return with an
emphasis on current distributions paid to shareholders. The Fund seeks to
achieve its objective by investing primarily in securities of companies engaged
in the energy infrastructure sector. Energy infrastructure companies principally
include publicly-traded master limited partnerships and limited liability
companies taxed as partnerships ("MLPs"), MLP affiliates, Canadian income
equities, pipeline companies, utilities, and other companies that derive at
least 50% of their revenues from operating or providing services in support of
infrastructure assets such as pipelines, power transmission and petroleum and
natural gas storage in the petroleum, natural gas and power generation
industries (collectively, "Energy Infrastructure Companies"). For purposes of
the Fund's investment objective, total return includes capital appreciation of,
and all distributions received from, securities in which the Fund will invest,
taking into account the varying tax characteristics of such securities. There
can be no assurance that the Fund will achieve its investment objective. The
Fund may not be appropriate for all investors.

                       2. SIGNIFICANT ACCOUNTING POLICIES

The following is a summary of significant accounting policies consistently
followed by the Fund in the preparation of its financial statements. The
preparation of financial statements in accordance with accounting principles
generally accepted in the United States of America ("U.S. GAAP") requires
management to make estimates and assumptions that affect the reported amounts
and disclosures in the financial statements. Actual results could differ from
those estimates.

A. PORTFOLIO VALUATION:

The net asset value ("NAV") of the Common Shares of the Fund is determined daily
as of the close of regular trading on the NYSE, normally 4:00 p.m. Eastern time,
on each day the NYSE is open for trading. If the NYSE closes early on a
valuation day, the NAV is determined as of that time. Foreign securities are
priced using data reflecting the earlier closing of the principal markets for
those securities. The NAV per Common Share is calculated by dividing the value
of all assets of the Fund (including accrued interest and dividends), less all
liabilities (including accrued expenses, dividends declared but unpaid and any
borrowings of the Fund) by the total number of Common Shares outstanding.

The Fund's investments are valued daily at market value or, in the absence of
market value with respect to any portfolio securities, at fair value in
accordance with valuation procedures adopted by the Fund's Board of Trustees,
and in accordance with provisions of the 1940 Act. Market quotations and prices
used to value the Fund's investments are primarily obtained from third party
pricing services. The Fund's investments are valued as follows:

       Common stocks, MLPs and other equity securities listed on any national or
       foreign exchange (excluding The NASDAQ(R) Stock Market, LLC ("NASDAQ")
       and the London Stock Exchange Alternative Investment Market ("AIM")) are
       valued at the last sale price on the exchange on which they are
       principally traded or, for NASDAQ and AIM securities, the official
       closing price. Securities traded on more than one securities exchange are
       valued at the last sale price or official closing price, as applicable,
       at the close of the securities exchange representing the principal market
       for such securities.

       Exchange-traded options contracts are valued at the closing price in the
       market where such contracts are principally traded. If no closing price
       is available, exchange-traded options contracts are valued at the mean of
       their most recent bid and asked price, if available, and otherwise at
       their closing bid price. Over-the-counter options contracts are valued at
       the mean of their most recent bid and asked price, if available, and
       otherwise at their closing bid price.

       Securities traded in the over-the-counter market are valued at the mean
       of their most recent bid and asked price, if available, and otherwise at
       their closing bid price.

       Swaps are valued utilizing quotations provided by a third party pricing
       service or, if the pricing service does not provide a value, by quotes
       provided by the selling dealer or financial institution.

       Short-term investments that mature in less than 60 days when purchased
       are valued at amortized cost.

Certain securities may not be able to be priced by pre-established pricing
methods. Such securities may be valued by the Fund's Board of Trustees or its
delegate at fair value. These securities generally include, but are not limited
to, restricted securities (securities which may not be publicly sold without
registration under the Securities Act of 1933, as amended) for which a pricing
service is unable to provide a market price; securities whose trading has been
formally suspended; a security whose market price is not available from a
pre-established pricing source; a security with respect to which an event has
occurred that is likely to materially affect the value of the security after the
market has closed but before the calculation of the Fund's NAV or make it
difficult or impossible to obtain a reliable market quotation; and a security
whose price, as provided by the pricing service, does not reflect the security's
"fair value." As a general principle, the current "fair value" of a security
would appear to be the amount which the owner might reasonably expect to receive
for the security upon its current sale. The use of fair value prices


                                                                         Page 15





--------------------------------------------------------------------------------
NOTES TO FINANCIAL STATEMENTS (CONTINUED)
--------------------------------------------------------------------------------

                  FIRST TRUST ENERGY INFRASTRUCTURE FUND (FIF)
                            MAY 31, 2014 (UNAUDITED)


by the Fund generally results in prices used by the Fund that may differ from
current market quotations or official closing prices on the applicable exchange.
A variety of factors may be considered in determining the fair value of such
securities, including, but not limited to, the following:

      1)    the type of security;
      2)    the size of the holding;
      3)    the initial cost of the security;
      4)    transactions in comparable securities;
      5)    price quotes from dealers and/or pricing services;
      6)    relationships among various securities;
      7)    information obtained by contacting the issuer, analysts, or the
            appropriate stock exchange;
      8)    an analysis of the issuer's financial statements; and
      9)    the existence of merger proposals or tender offers that might affect
            the value of the security.

If the securities in question are foreign securities, the following additional
information may be considered:

      1)    the value of similar foreign securities traded on other foreign
            markets;
      2)    ADR trading of similar securities;
      3)    closed-end fund trading of similar securities;
      4)    foreign currency exchange activity;
      5)    the trading prices of financial products that are tied to baskets of
            foreign securities;
      6)    factors relating to the event that precipitated the pricing problem;
      7)    whether the event is likely to recur; and
      8)    whether the effects of the event are isolated or whether they affect
            entire markets, countries or regions.

The Fund is subject to fair value accounting standards that define fair value,
establish the framework for measuring fair value and provide a three-level
hierarchy for fair valuation based upon the inputs to the valuation as of the
measurement date. The three levels of the fair value hierarchy are as follows:

      o     Level 1 - Level 1 inputs are quoted prices in active markets for
            identical investments. An active market is a market in which
            transactions for the investment occur with sufficient frequency and
            volume to provide pricing information on an ongoing basis.

      o     Level 2 - Level 2 inputs are observable inputs, either directly or
            indirectly, and include the following:

            o     Quoted prices for similar investments in active markets.

            o     Quoted prices for identical or similar investments in markets
                  that are non-active. A non-active market is a market where
                  there are few transactions for the investment, the prices are
                  not current, or price quotations vary substantially either
                  over time or among market makers, or in which little
                  information is released publicly.

            o     Inputs other than quoted prices that are observable for the
                  investment (for example, interest rates and yield curves
                  observable at commonly quoted intervals, volatilities,
                  prepayment speeds, loss severities, credit risks, and default
                  rates).

            o     Inputs that are derived principally from or corroborated by
                  observable market data by correlation or other means.

      o     Level 3 - Level 3 inputs are unobservable inputs. Unobservable
            inputs may reflect the reporting entity's own assumptions about the
            assumptions that market participants would use in pricing the
            investment.

The inputs or methodology used for valuing investments are not necessarily an
indication of the risk associated with investing in those investments. A summary
of the inputs used to value the Fund's investments as of May 31, 2014, is
included with the Fund's Portfolio of Investments.

B. OPTION CONTRACTS:

The Fund is subject to equity price risk in the normal course of pursuing its
investment objective and may write (sell) options to hedge against changes in
the value of equities. Also, the Fund seeks to generate additional income, in
the form of premiums received, from writing (selling) the options. The Fund may
write (sell) covered call or put options ("options") on all or a portion of the
common stock and MLPs held in the Fund's portfolio as determined to be
appropriate by Energy Income Partners, LLC ("EIP" or the "Sub-Advisor"). The
number of options the Fund can write (sell) is limited by the amount of common
stock and MLPs the Fund holds in its portfolio. The Fund will not write (sell)
"naked" or uncovered options. When the Fund writes (sells) an option, an amount
equal to the premium received by the Fund is included in "Options written, at
value" on the Fund's Statement of Assets and Liabilities. Options are
marked-to-market daily and their value will be affected by changes in the value
and dividend rates of the underlying equity securities, changes in interest
rates, changes in the actual or perceived volatility of the securities markets
and the underlying equity securities and the remaining time to the options'
expiration. The value of options may also be adversely affected if the market
for the options becomes less liquid or trading volume diminishes.

Options the Fund writes (sells) will either be exercised, expire or be cancelled
pursuant to a closing transaction. If the price of the underlying security
exceeds the option's exercise price, it is likely that the option holder will
exercise the option. If an option written (sold) by the Fund is exercised, the
Fund would be obligated to deliver the underlying security to the option holder
upon payment of the strike price. In this case, the option premium received by
the Fund will be added to the amount realized on the sale of the underlying
security for purposes of determining


Page 16





--------------------------------------------------------------------------------
NOTES TO FINANCIAL STATEMENTS (CONTINUED)
--------------------------------------------------------------------------------

                  FIRST TRUST ENERGY INFRASTRUCTURE FUND (FIF)
                            MAY 31, 2014 (UNAUDITED)


gain or loss. If the price of the underlying security is less than the option's
strike price, the option will likely expire without being exercised. The option
premium received by the Fund will, in this case, be treated as short-term
capital gain on the expiration date of the option. The Fund may also elect to
close out its position in an option prior to its expiration by purchasing an
option of the same series as the option written (sold) by the Fund. Gain or loss
on options is presented separately as "Net realized gain (loss) on written
options" on the Statement of Operations.

The options that the Fund writes (sells) give the option holder the right, but
not the obligation, to purchase a security from the Fund at the strike price on
or prior to the option's expiration date. The ability to successfully implement
the writing (selling) of covered call options depends on the ability of the
Sub-Advisor to predict pertinent market movements, which cannot be assured.
Thus, the use of options may require the Fund to sell portfolio securities at
inopportune times or for prices other than current market value, which may limit
the amount of appreciation the Fund can realize on an investment, or may cause
the Fund to hold a security that it might otherwise sell. As the writer (seller)
of a covered option, the Fund foregoes, during the option's life, the
opportunity to profit from increases in the market value of the security
covering the option above the sum of the premium and the strike price of the
option, but has retained the risk of loss should the price of the underlying
security decline. The writer (seller) of an option has no control over the time
when it may be required to fulfill its obligation as a writer (seller) of the
option. Once an option writer (seller) has received an exercise notice, it
cannot effect a closing purchase transaction in order to terminate its
obligation under the option and must deliver the underlying security to the
option holder at the exercise price.

Over-the-counter options have the risk of the potential inability of
counterparties to meet the terms of their contracts. The Fund's maximum equity
price risk for purchased options is limited to the premium initially paid. In
addition, certain risks may arise upon entering into option contracts including
the risk that an illiquid secondary market will limit the Fund's ability to
close out an option contract prior to the expiration date and that a change in
the value of the option contract may not correlate exactly with changes in the
value of the securities hedged.

C. SWAP AGREEMENTS:

The Fund may enter into total return equity swap and interest rate swap
agreements. A swap is a financial instrument that typically involves the
exchange of cash flows between two parties ("Counterparties") on specified dates
(settlement dates) where the cash flows are based on agreed-upon prices, rates,
etc. Swap agreements are individually negotiated and involve the risk of the
potential inability of the Counterparties to meet the terms of the agreement. In
connection with these agreements, cash and securities may be identified as
collateral in accordance with the terms of the respective swap agreements to
provide assets of value and recourse in the event of default under the swap
agreement or bankruptcy/insolvency of a party to the swap agreement. In the
event of a default by the Counterparty, the Fund will seek withdrawal of this
collateral and may incur certain costs exercising its right with respect to the
collateral. If a Counterparty becomes bankrupt or otherwise fails to perform its
obligations due to financial difficulties, the Fund may experience significant
delays in obtaining any recovery in a bankruptcy or other reorganization
proceeding. The Fund may obtain only limited recovery or may obtain no recovery
in such circumstances.

Swap agreements may increase or decrease the overall volatility of the
investments of the Fund. The performance of swap agreements may be affected by a
change in the specific interest rate, security, currency, or other factors that
determine the amounts of payments due to and from the Fund. The Fund's maximum
equity price risk to meet its future payments under swap agreements outstanding
at May 31, 2014 is equal to the total notional amount as shown on the Portfolio
of Investments. The notional amount represents the U.S. dollar value of the
contract as of the day of the opening transaction or contract reset. When the
Fund enters into a swap agreement, any premium paid is included in "Swap
contracts, at value" on the Statement of Assets and Liabilities.

An interest rate swap agreement involves the Fund's agreement to exchange a
stream of interest payments for another party's stream of cash flows. Interest
rate swaps do not involve the delivery of securities or other underlying assets
or principal. Accordingly, the risk of loss with respect to interest rate swaps
is limited to the net amount of interest payments that the Fund is contractually
obligated to make.

D. SECURITIES TRANSACTIONS AND INVESTMENT INCOME:

Securities transactions are recorded as of the trade date. Realized gains and
losses from securities transactions are recorded on the identified cost basis.
Dividend income is recorded on the ex-dividend date. Interest income is recorded
daily on the accrual basis, including amortization of premiums and accretion of
discounts. The Fund will rely to some extent on information provided by the
MLPs, which is not necessarily timely, to estimate taxable income allocable to
the MLP units held in the Fund's portfolio.

Distributions received from the Fund's investments in MLPs generally are
comprised of return of capital and investment income. The Fund records estimated
return of capital and investment income based on historical information
available from each MLP. These estimates may subsequently be revised based on
information received from the MLPs after their tax reporting periods are
concluded. For the six months ended May 31, 2014, distributions of $3,772,153
received from MLPs have been reclassified as return of capital. The cost basis
of applicable MLPs has been reduced accordingly.

E. DIVIDENDS AND DISTRIBUTIONS TO SHAREHOLDERS:

The Fund will distribute to holders of its Common Shares monthly dividends of
all or a portion of its net income after the payment of interest in connection
with leverage, if any. Distributions of any long-term capital gains earned by
the Fund are distributed at least annually. Distributions will automatically be
reinvested into additional Common Shares pursuant to the Fund's Dividend
Reinvestment Plan unless cash distributions are elected by the shareholder.


                                                                         Page 17





--------------------------------------------------------------------------------
NOTES TO FINANCIAL STATEMENTS (CONTINUED)
--------------------------------------------------------------------------------

                  FIRST TRUST ENERGY INFRASTRUCTURE FUND (FIF)
                            MAY 31, 2014 (UNAUDITED)


Distributions from income and capital gains are determined in accordance with
income tax regulations, which may differ from U.S. GAAP. Certain capital
accounts in the financial statements are periodically adjusted for permanent
differences in order to reflect their tax character. These permanent differences
are primarily due to the varying treatment of income and gain/loss on portfolio
securities held by the Fund and have no impact on net assets or NAV per share.
Temporary differences, which arise from recognizing certain items of income,
expense and gain/loss in different periods for financial statement and tax
purposes, will reverse at some point in the future.

The tax character of distributions paid during the fiscal year ended November
30, 2013 was as follows:


Distributions paid from:
Ordinary income...................................   $  36,268,003
Capital gain......................................      35,321,879
Return of capital.................................         918,918

As of November 30, 2013, the components of distributable earnings and net assets
on a tax basis were as follows:


Undistributed ordinary income.....................   $          --
Undistributed capital gains.......................              --
                                                     -------------
Total undistributed earnings......................              --
Accumulated capital and other losses..............              --
Net unrealized appreciation (depreciation)........      57,561,098
                                                     -------------
Total accumulated earnings (losses)...............      57,561,098
Other   ..........................................         (54,252)
Paid-in capital...................................     333,829,544
                                                     -------------
Net assets........................................   $ 391,336,390
                                                     =============

F. INCOME TAXES:

The Fund intends to continue to qualify as a regulated investment company by
complying with the requirements under Subchapter M of the Internal Revenue Code
of 1986, as amended, which includes distributing substantially all of its net
investment income and net realized gains to shareholders. Accordingly, no
provision has been made for federal or state income taxes. However, due to the
timing and amount of distributions, the Fund may be subject to an excise tax of
4% of the amount by which approximately 98% of the Fund's taxable income exceeds
the distributions from such taxable income for the calendar year.

The Fund intends to utilize provisions of the federal income tax laws, which
allow it to carry a realized capital loss forward indefinitely following the
year of the loss and offset such loss against any future realized capital gains.
The Fund is subject to certain limitations under U.S. tax rules on the use of
capital loss carryforwards and net unrealized built-in losses. These limitations
apply when there has been a 50% change in ownership. At November 30, 2013, the
Fund had no capital loss carryforwards for federal income tax purposes.

Certain losses realized during the current fiscal year may be deferred and
treated as occurring on the first day of the following fiscal year for federal
income tax purposes. For the fiscal year ended November 30, 2013, the Fund had
no qualified late year losses.

The Fund is subject to accounting standards that establish a minimum threshold
for recognizing, and a system for measuring, the benefits of a tax position
taken or expected to be taken in a tax return. Taxable years ending 2011, 2012
and 2013 remain open to federal and state audit. As of May 31, 2014, management
has evaluated the application of these standards to the Fund and has determined
that no provision for income tax is required in the Fund's financial statements
for uncertain tax positions.

G. EXPENSES:

The Fund will pay all expenses directly related to its operations.

H. FOREIGN CURRENCY:

The books and records of the Fund are maintained in U.S. dollars. Foreign
currencies, investments and other assets and liabilities are translated into
U.S. dollars at the exchange rates prevailing at the end of the period.
Purchases and sales of investment securities and items of income and expense are
translated on the respective dates of such transactions. Unrealized gains and
losses on assets and liabilities, other than investments in securities, which
result from changes in foreign currency exchange rates have been included in
"Net change in unrealized appreciation (depreciation) on foreign currency
translation" on the Statement of Operations. Unrealized gains and losses on
investments in securities which result from changes in foreign exchange rates
are included with fluctuations arising from changes in market price and are
shown in "Net change in unrealized appreciation (depreciation) on investments"
on the Statement of Operations. Net realized foreign currency gains and losses
include the effect of changes in exchange rates between trade date and
settlement date on investment security transactions, foreign currency
transactions and interest and dividends received. The portion of foreign
currency gains and losses related to fluctuation in exchange rates between the
initial purchase settlement date and subsequent sale trade date is included in
"Net realized gain (loss) on investments" on the Statement of Operations.


Page 18





--------------------------------------------------------------------------------
NOTES TO FINANCIAL STATEMENTS (CONTINUED)
--------------------------------------------------------------------------------

                  FIRST TRUST ENERGY INFRASTRUCTURE FUND (FIF)
                            MAY 31, 2014 (UNAUDITED)


 3. INVESTMENT ADVISORY FEE, AFFILIATED TRANSACTIONS AND OTHER FEE ARRANGEMENTS

First Trust Advisors L.P. ("First Trust"), the investment advisor to the Fund,
is a limited partnership with one limited partner, Grace Partners of DuPage
L.P., and one general partner, The Charger Corporation. The Charger Corporation
is an Illinois corporation controlled by James A. Bowen, Chief Executive Officer
of First Trust. First Trust is responsible for the ongoing monitoring of the
Fund's investment portfolio, managing the Fund's business affairs and providing
certain administrative services necessary for the management of the Fund. For
these investment management services, First Trust is entitled to a monthly fee
calculated at an annual rate of 1.00% of the Fund's Managed Assets (the average
daily total asset value of the Fund minus the sum of the Fund's liabilities
other than the principal amount of borrowings). First Trust also provides fund
reporting services to the Fund for a flat annual fee in the amount of $9,250.

EIP serves as the Fund's Sub-Advisor and manages the Fund's portfolio subject to
First Trust's supervision. The Sub-Advisor receives a monthly Sub-Advisory fee
calculated at an annual rate of 0.50% of the Fund's Managed Assets that is paid
by First Trust out of its investment advisory fee.

During the fiscal year ended November 30, 2013, the Fund received reimbursements
from the Sub-Advisor of $5,421 in connection with trade errors.

First Trust Capital Partners, LLC ("FTCP"), an affiliate of First Trust, owns,
through a wholly-owned subsidiary, a 15% ownership interest in each of EIP and
EIP Partners, LLC, an affiliate of EIP. In addition, as of March 27, 2014, FTCP,
through a wholly-owned subsidiary, purchased a preferred interest in EIP. The
preferred interest is non-voting and does not share in the profits or losses of
EIP. EIP may buy back any or all of FTCP's preferred interest at any time and
FTCP may sell back to EIP up to 50% of its preferred interest on or after
September 25, 2015, and any or all of its preferred interest after March 27,
2017.

BNY Mellon Investment Servicing (US) Inc. ("BNYM IS") serves as the Fund's
administrator, fund accountant and transfer agent in accordance with certain fee
arrangements. As administrator and fund accountant, BNYM IS is responsible for
providing certain administrative and accounting services to the Fund, including
maintaining the Fund's books of account, records of the Fund's securities
transactions, and certain other books and records. As transfer agent, BNYM IS is
responsible for maintaining shareholder records for the Fund. The Bank of New
York Mellon ("BNYM") serves as the Fund's custodian in accordance with certain
fee arrangements. As custodian, BNYM is responsible for custody of the Fund's
assets.

Each Trustee who is not an officer or employee of First Trust, any Sub-Advisor
or any of their affiliates ("Independent Trustees") is paid a fixed annual
retainer that is allocated pro rata among each fund in the First Trust Fund
Complex based on net assets. Each Independent Trustee is also paid an annual per
fund fee that varies based on whether the fund is a closed-end or other actively
managed fund, or is an index fund.

Additionally, the Lead Independent Trustee and the Chairmen of the Audit
Committee, Nominating and Governance Committee and Valuation Committee are paid
annual fees to serve in such capacities, with such compensation allocated pro
rata among each fund in the First Trust Fund Complex based on net assets.
Trustees are reimbursed for travel and out-of-pocket expenses in connection with
all meetings. The Lead Independent Trustee and Committee Chairmen rotate every
three years. The officers and "Interested" Trustee receive no compensation from
the Fund for acting in such capacities.

                      4. PURCHASES AND SALES OF SECURITIES

Cost of purchases and proceeds from sales of investments, excluding short-term
investments, for the six months ended May 31, 2014, were $94,227,699 and
$109,538,603, respectively.

                          5. DERIVATIVES TRANSACTIONS

Written option activity for the Fund was as follows:

                                                  NUMBER
                                                    OF
WRITTEN OPTIONS                                  CONTRACTS      PREMIUMS
--------------------------------------------------------------------------------
Options outstanding at November 30, 2013...         32,393    $  1,571,359
Options Written............................         52,156       2,642,536
Options Expired............................        (28,595)     (1,156,986)
Options Exercised..........................        (14,603)       (748,962)
Options Closed.............................         (8,640)       (660,165)
                                                 ---------    ------------
Options outstanding at May 31, 2014........         32,711    $  1,647,782
                                                 =========    ============


                                                                         Page 19





--------------------------------------------------------------------------------
NOTES TO FINANCIAL STATEMENTS (CONTINUED)
--------------------------------------------------------------------------------

                  FIRST TRUST ENERGY INFRASTRUCTURE FUND (FIF)
                            MAY 31, 2014 (UNAUDITED)


The following table presents the types of derivatives held by the Fund at May
31, 2014, the location of these instruments as presented on the Statement of
Assets and Liabilities and the primary underlying risk exposure.



                                                 ASSET DERIVATIVES                       LIABILITY DERIVATIVES
                                      ----------------------------------------  ----------------------------------------
                     DERIVATIVE        STATEMENT OF ASSETS AND                   STATEMENT OF ASSETS AND
RISK EXPOSURE        INSTRUMENT         LIABILITIES LOCATION      FAIR VALUE      LIABILITIES LOCATION      FAIR VALUE
-------------   --------------------  -------------------------  -------------  -------------------------  -------------
                                                                                            
Equity Risk     Written Options       Options written, at value  $     --       Options written, at value  $   3,011,785
Interest Rate   Interest Rate
Risk            Swap Agreements       Swap contracts, at value         --       Swap contracts, at value         685,474


The following table presents the amount of net realized gain (loss) and change
in net unrealized appreciation (depreciation) recognized for the six months
ended May 31, 2014, on derivative instruments, as well as the primary underlying
risk exposure associated with each instrument.


STATEMENT OF OPERATIONS LOCATION
--------------------------------------------------------------------------------
EQUITY RISK EXPOSURE
Net realized gain (loss) on written options                          $  (642,929)
Net change in unrealized gain (loss) on written options transactions  (1,683,462)

INTEREST RATE RISK EXPOSURE
Net realized gain (loss) on swap contracts                              (359,465)
Net change in unrealized gain (loss) on swap contracts                  (404,619)


The average volume of interest rate swaps was $36,475,000 for the six months
ended May 31, 2014.

                                 6. BORROWINGS

The Fund has a committed facility agreement with The Bank of Nova Scotia
("Scotia") that has a maximum commitment amount of $165,000,000. The borrowing
rate under the facility is equal to the 1-month LIBOR plus 65 basis points. In
addition, under the facility, the Fund pays a commitment fee of 0.20% on the
undrawn amount of such facility. The average amount outstanding for the six
months ended May 31, 2014 was $145,900,000, with a weighted average interest
rate of 0.76%. As of May 31, 2014, the Fund had outstanding borrowings of
$145,900,000 under this committed facility agreement. The high and low annual
interest rates for the six months ended May 31, 2014, were 0.77% and 0.75%,
respectively. The interest rate at May 31, 2014 was 0.75%.

                               7. INDEMNIFICATION

The Fund has a variety of indemnification obligations under contracts with its
service providers. The Fund's maximum exposure under these arrangements is
unknown. However, the Fund has not had prior claims or losses pursuant to these
contracts and expects the risk of loss to be remote.

                         8. INDUSTRY CONCENTRATION RISK

The Fund invests at least 80% of its Managed Assets in securities issued by
Energy Infrastructure Companies. Given this industry concentration, the Fund is
more susceptible to adverse economic or regulatory occurrences affecting that
industry than an investment company that is not concentrated in a single
industry. Energy Infrastructure Company issuers may be subject to a variety of
factors that may adversely affect their business or operations, including high
interest costs in connection with capital construction programs, high leverage
costs associated with environmental and other regulations, the effects of
economic slowdown, surplus capacity, increased competition from other providers
of services, uncertainties concerning the availability of fuel at reasonable
prices, the effects of energy conservation policies and other factors.

                              9. SUBSEQUENT EVENTS

Management has evaluated the impact of all subsequent events to the Fund through
the date the financial statements were issued, and has determined that there
were the following subsequent events:

On June 19, 2014, the Fund declared a dividend of $0.11 per share to Common
Shareholders of record on July 3, 2014, payable July 15, 2014.

On July 21, 2014, the Fund declared a dividend of $0.11 per share to Common
Shareholders of record on August 5, 2014, payable August 15, 2014.


Page 20




--------------------------------------------------------------------------------
ADDITIONAL INFORMATION
--------------------------------------------------------------------------------

                 FIRST TRUST ENERGY INFRASTRUCTURE FUND (FIF)
                            MAY 31, 2014 (UNAUDITED)


                           DIVIDEND REINVESTMENT PLAN

If your Common Shares are registered directly with the Fund or if you hold your
Common Shares with a brokerage firm that participates in the Fund's Dividend
Reinvestment Plan (the "Plan"), unless you elect, by written notice to the Fund,
to receive cash distributions, all dividends, including any capital gain
distributions, on your Common Shares will be automatically reinvested by BNY
Mellon Investment Servicing (US) Inc. (the "Plan Agent"), in additional Common
Shares under the Plan. If you elect to receive cash distributions, you will
receive all distributions in cash paid by check mailed directly to you by the
Plan Agent, as the dividend paying agent.

If you decide to participate in the Plan, the number of Common Shares you will
receive will be determined as follows:

      (1)   If Common Shares are trading at or above net asset value ("NAV") at
            the time of valuation, the Fund will issue new shares at a price
            equal to the greater of (i) NAV per Common Share on that date or
            (ii) 95% of the market price on that date.

      (2)   If Common Shares are trading below NAV at the time of valuation, the
            Plan Agent will receive the dividend or distribution in cash and
            will purchase Common Shares in the open market, on the NYSE or
            elsewhere, for the participants' accounts. It is possible that the
            market price for the Common Shares may increase before the Plan
            Agent has completed its purchases. Therefore, the average purchase
            price per share paid by the Plan Agent may exceed the market price
            at the time of valuation, resulting in the purchase of fewer shares
            than if the dividend or distribution had been paid in Common Shares
            issued by the Fund. The Plan Agent will use all dividends and
            distributions received in cash to purchase Common Shares in the open
            market within 30 days of the valuation date except where temporary
            curtailment or suspension of purchases is necessary to comply with
            federal securities laws. Interest will not be paid on any uninvested
            cash payments.

You may elect to opt-out of or withdraw from the Plan at any time by giving
written notice to the Plan Agent, or by telephone at (866) 340-1104, in
accordance with such reasonable requirements as the Plan Agent and the Fund may
agree upon. If you withdraw or the Plan is terminated, you will receive a
certificate for each whole share in your account under the Plan, and you will
receive a cash payment for any fraction of a share in your account. If you wish,
the Plan Agent will sell your shares and send you the proceeds, minus brokerage
commissions.

The Plan Agent maintains all Common Shareholders' accounts in the Plan and gives
written confirmation of all transactions in the accounts, including information
you may need for tax records. Common Shares in your account will be held by the
Plan Agent in non-certificated form. The Plan Agent will forward to each
participant any proxy solicitation material and will vote any shares so held
only in accordance with proxies returned to the Fund. Any proxy you receive will
include all Common Shares you have received under the Plan.

There is no brokerage charge for reinvestment of your dividends or distributions
in Common Shares. However, all participants will pay a pro rata share of
brokerage commissions incurred by the Plan Agent when it makes open market
purchases.

Automatically reinvesting dividends and distributions does not mean that you do
not have to pay income taxes due upon receiving dividends and distributions.
Capital gains and income are realized although cash is not received by you.
Consult your financial advisor for more information.

If you hold your Common Shares with a brokerage firm that does not participate
in the Plan, you will not be able to participate in the Plan and any dividend
reinvestment may be effected on different terms than those described above.

The Fund reserves the right to amend or terminate the Plan if in the judgment of
the Board of Trustees the change is warranted. There is no direct service charge
to participants in the Plan; however, the Fund reserves the right to amend the
Plan to include a service charge payable by the participants. Additional
information about the Plan may be obtained by writing BNY Mellon Investment
Servicing (US) Inc., 301 Bellevue Parkway, Wilmington, Delaware 19809.

                      PROXY VOTING POLICIES AND PROCEDURES

A description of the policies and procedures that the Fund uses to determine how
to vote proxies and information on how the Fund voted proxies relating to
portfolio investments during the most recent 12-month period ended June 30 is
available (1) without charge, upon request, by calling (800) 988-5891; (2) on
the Fund's website located at http://www.ftportfolios.com; and (3) on the
Securities and Exchange Commission's ("SEC") website located at
http://www.sec.gov.


Page 21





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ADDITIONAL INFORMATION (CONTINUED)
--------------------------------------------------------------------------------

                  FIRST TRUST ENERGY INFRASTRUCTURE FUND (FIF)
                            MAY 31, 2014 (UNAUDITED)


                               PORTFOLIO HOLDINGS

The Fund files its complete schedule of portfolio holdings with the SEC for the
first and third quarters of each fiscal year on Form N-Q. The Fund's Forms N-Q
are available (1) by calling (800) 988-5891; (2) on the Fund's website located
at http://www.ftportfolios.com; (3) on the SEC's website at http://www.sec.gov;
and (4) for review and copying at the SEC's Public Reference Room ("PRR") in
Washington, DC. Information regarding the operation of the PRR may be obtained
by calling (800) SEC-0330.

                SUBMISSION OF MATTERS TO A VOTE OF SHAREHOLDERS

The Joint Annual Meeting of Shareholders of the Common Shares of Macquarie/First
Trust Global Infrastructure/Utilities Dividend & Income Fund, First Trust Energy
Income and Growth Fund, First Trust Enhanced Equity Income Fund, First
Trust/Aberdeen Global Opportunity Income Fund, First Trust Mortgage Income Fund,
First Trust Strategic High Income Fund II, First Trust/Aberdeen Emerging
Opportunity Fund, First Trust Specialty Finance and Financial Opportunities
Fund, First Trust Dividend and Income Fund, First Trust High Income Long/Short
Fund, First Trust Energy Infrastructure Fund, First Trust MLP and Energy Income
Fund and First Trust Intermediate Duration Preferred & Income Fund was held on
April 23, 2014 (the "Annual Meeting"). At the Annual Meeting, Robert F. Keith
was elected by the Common Shareholders of the First Trust Energy Infrastructure
Fund as a Class I Trustee for a three-year term expiring at the Fund's annual
meeting of shareholders in 2017. The number of votes cast for Mr. Keith was
15,622,207, the number of votes against was 247,925 and the number of broker
non-votes was 1,680,104. James A. Bowen, Richard E. Erickson, Thomas R. Kadlec
and Niel B. Nielson are the other current and continuing Trustees.

                              RISK CONSIDERATIONS

Risks are inherent in all investing. The following summarizes some, but not all,
of the risks that should be considered for the Fund. For additional information
about the risks associated with investing in the Fund, please see the Fund's
prospectus and statement of additional information, as well as other Fund
regulatory filings.

CURRENCY RISK: The value of securities denominated or quoted in foreign
currencies may be adversely affected by fluctuations in the relative currency
exchange rates and by exchange control regulations. The Fund's investment
performance may be negatively affected by a devaluation of a currency in which
the Fund's investments are denominated or quoted. Further, the Fund's investment
performance may be significantly affected, either positively or negatively, by
currency exchange rates because the U.S. dollar value of securities denominated
or quoted in another currency will increase or decrease in response to changes
in the value of such currency in relation to the U.S. dollar. While certain of
the Fund's non-U.S. dollar-denominated securities may be hedged into U.S.
dollars, hedging may not alleviate all currency risks.

INVESTMENT AND MARKET RISK: An investment in the Fund's Common Shares is subject
to investment risk, including the possible loss of the entire principal
invested. An investment in Common Shares represents an indirect investment in
the securities owned by the Fund. The value of these securities, like other
market investments, may move up or down, sometimes rapidly and unpredictably.
Common Shares at any point in time may be worth less than the original
investment, even after taking into account the reinvestment of Fund dividends
and distributions. Security prices can fluctuate for several reasons including
the general condition of the securities markets, or when political or economic
events affecting the issuers occur. When the Advisor or Sub-Advisor determines
that it is temporarily unable to follow the Fund's investment strategy or that
it is impractical to do so (such as when a market disruption event has occurred
and trading in the securities is extremely limited or absent), the Fund may take
temporary defensive positions.

LEVERAGE RISK: The use of leverage results in additional risks and can magnify
the effect of any losses. The funds borrowed pursuant to a leverage borrowing
program constitute a substantial lien and burden by reason of their prior claim
against the income of the Fund and against the net assets of the Fund in
liquidation. If the Fund is not in compliance with certain credit facility
provisions, the Fund may not be permitted to declare dividends or other
distributions.

MLP RISK: An investment in MLP units involves risks which differ from an
investment in common stock of a corporation. Holders of MLP units have limited
control and voting rights on matters affecting the partnership. In addition,
there are certain tax risks associated with an investment in MLP units and
conflicts of interest exist between common unit holders and the general partner,
including those arising from incentive distribution payments.

NON-U.S. RISK: The Fund may invest a portion of its assets in the equity
securities of issuers domiciled in jurisdictions other than the U.S. Investments
in the securities and instruments of non-U.S. issuers involve certain
considerations and risks not ordinarily associated with investments in
securities and instruments of U.S. issuers. Non-U.S. companies are not generally
subject to uniform accounting, auditing and financial standards and requirements
comparable to those applicable to U.S. companies. Non-U.S. securities exchanges,
brokers and listed


Page 22




--------------------------------------------------------------------------------
ADDITIONAL INFORMATION (CONTINUED)
--------------------------------------------------------------------------------

                  FIRST TRUST ENERGY INFRASTRUCTURE FUND (FIF)
                            MAY 31, 2014 (UNAUDITED)


companies may be subject to less government supervision and regulation than
exists in the United States. Dividend and interest income may be subject to
withholding and other non-U.S. taxes, which may adversely affect the net return
on such investments. A related risk is that there may be difficulty in obtaining
or enforcing a court judgment abroad.

QUALIFIED DIVIDEND INCOME TAX RISK: There can be no assurance as to what portion
of the distributions paid to the Fund's Common Shareholders will consist of
tax-advantaged qualified dividend income. Certain distributions designated by
the Fund as derived from qualified dividend income will be taxed in the hands of
non-corporate Common Shareholders at the rates applicable to long-term capital
gains, provided certain holding period and other requirements are satisfied by
both the Fund and the Common Shareholders. Additional requirements apply in
determining whether distributions by foreign issuers should be regarded as
qualified dividend income. Certain investment strategies of the Fund will limit
the Fund's ability to meet these requirements and consequently will limit the
amount of qualified dividend income received and distributed by the Fund. A
change in the favorable provisions of the federal tax laws with respect to
qualified dividends may result in a widespread reduction in announced dividends
and may adversely impact the valuation of the shares of dividend-paying
companies.

RESTRICTED SECURITIES RISK: The Fund may invest in unregistered or otherwise
restricted securities. The term "restricted securities" refers to securities
that are unregistered or are held by control persons of the issuer and
securities that are subject to contractual restrictions on their resale. As a
result, restricted securities may be more difficult to value and the Fund may
have difficulty disposing of such assets either in a timely manner or for a
reasonable price. In order to dispose of an unregistered security, the Fund,
where it has contractual rights to do so, may have to cause such security to be
registered. A considerable period may elapse between the time the decision is
made to sell the security and the time the security is registered so that the
Fund could sell it. Contractual restrictions on the resale of securities vary in
length and scope and are generally the result of a negotiation between the
issuer and acquirer of the securities. The Fund would, in either case, bear
market risks during that period.


                                                                         Page 23





                      This Page Left Blank Intentionally.





FIRST TRUST

INVESTMENT ADVISOR
First Trust Advisors L.P.
120 E. Liberty Drive, Suite 400
Wheaton, IL 60187

INVESTMENT SUB-ADVISOR
Energy Income Partners, LLC
49 Riverside Avenue
Westport, CT 06880

ADMINISTRATOR,
FUND ACCOUNTANT &
TRANSFER AGENT
BNY Mellon Investment Servicing (US) Inc.
301 Bellevue Parkway
Wilmington, DE 19809

CUSTODIAN
The Bank of New York Mellon
101 Barclay Street, 20th Floor
New York, NY 10286

INDEPENDENT REGISTERED
PUBLIC ACCOUNTING FIRM
Deloitte & Touche LLP
111 S. Wacker Drive
Chicago, IL 60606

LEGAL COUNSEL
Chapman and Cutler LLP
111 W. Monroe Street
Chicago, IL 60603





[BLANK BACK COVER]






ITEM 2. CODE OF ETHICS.

Not applicable.


ITEM 3. AUDIT COMMITTEE FINANCIAL EXPERT.

Not applicable.


ITEM 4. PRINCIPAL ACCOUNTANT FEES AND SERVICES.

Not applicable.


ITEM 5. AUDIT COMMITTEE OF LISTED REGISTRANTS.

Not applicable.


ITEM 6. INVESTMENTS.

(a)   Schedule of Investments in securities of unaffiliated issuers as of the
      close of the reporting period is included as part of the report to
      shareholders filed under Item 1 of this form.

(b)   Not applicable.


ITEM 7. DISCLOSURE OF PROXY VOTING POLICIES AND PROCEDURES FOR CLOSED-END
MANAGEMENT INVESTMENT COMPANIES.

Not applicable.


ITEM 8. PORTFOLIO MANAGERS OF CLOSED-END MANAGEMENT INVESTMENT COMPANIES.

(a)   Not applicable.

(b)   There has been no change, as of the date of this filing, in any of the
      portfolio managers identified in response to paragraph (a)(1) of this Item
      in the registrant's most recently filed annual report on Form N-CSR.


ITEM 9. PURCHASES OF EQUITY SECURITIES BY CLOSED-END MANAGEMENT INVESTMENT
COMPANY AND AFFILIATED PURCHASERS.

Not applicable.


ITEM 10. SUBMISSION OF MATTERS TO A VOTE OF SECURITY HOLDERS.

There have been no material changes to the procedures by which the shareholders
may recommend nominees to the Registrant's board of trustees, where those
changes were implemented after the Registrant last provided disclosure in
response to the requirements of Item 407(c)(2)(iv) of Regulation S-K (17 CFR
229.407) (as required by Item 22(b)(15) of Schedule 14A (17 CFR 240.14a-101)),
or this Item.


ITEM 11. CONTROLS AND PROCEDURES.

(a)   The Registrant's principal executive and principal financial officers, or
      persons performing similar functions, have concluded that the Registrant's
      disclosure controls and procedures (as defined in Rule 30a-3(c) under the
      Investment Company Act of 1940, as amended (the "1940 Act") (17 CFR
      270.30a-3(c))) are effective, as of a date within 90 days of the filing
      date of the report that includes the disclosure required by this
      paragraph, based on their evaluation of these controls and procedures
      required by Rule 30a-3(b) under the 1940 Act (17 CFR 270.30a-3(b)) and
      Rules 13a-15(b) or 15d-15(b) under the Securities Exchange Act of 1934, as
      amended (17 CFR 240.13a-15(b) or 240.15d-15(b)).

(b)   There were no changes in the Registrant's internal control over financial
      reporting (as defined in Rule 30a-3(d) under the 1940 Act (17 CFR
      270.30a-3(d)) that occurred during the Registrant's second fiscal quarter
      of the period covered by this report that has materially affected, or is
      reasonably likely to materially affect, the Registrant's internal control
      over financial reporting.


ITEM 12. EXHIBITS.

(a)(1) Not applicable.

(a)(2) Certifications pursuant to Rule 30a-2(a) under the 1940 Act and Section
       302 of the Sarbanes-Oxley Act of 2002 are attached hereto.

(a)(3) Not applicable.

(b)    Certifications pursuant to Rule 30a-2(b) under the 1940 Act and Section
       906 of the Sarbanes- Oxley Act of 2002 are attached hereto.


                                   SIGNATURES

Pursuant to the requirements of the Securities Exchange Act of 1934 and the
Investment Company Act of 1940, the registrant has duly caused this report to be
signed on its behalf by the undersigned, thereunto duly authorized.

(registrant)         First Trust Energy Infrastructure Fund
             ------------------------------------------------------

By (Signature and Title)*               /s/ Mark R. Bradley
                                        ----------------------------------------
                                        Mark R. Bradley, President and
                                        Chief Executive Officer
                                        (principal executive officer)

Date: July 21, 2014
     -----------------

Pursuant to the requirements of the Securities Exchange Act of 1934 and the
Investment Company Act of 1940, this report has been signed below by the
following persons on behalf of the registrant and in the capacities and on the
dates indicated.

By (Signature and Title)*               /s/ Mark R. Bradley
                                        ----------------------------------------
                                        Mark R. Bradley, President and
                                        Chief Executive Officer
                                        (principal executive officer)

Date: July 21, 2014
     -----------------

By (Signature and Title)*               /s/ James M. Dykas
                                        ----------------------------------------
                                        James M. Dykas, Treasurer,
                                        Chief Financial Officer and
                                        Chief Accounting Officer
                                        (principal financial officer)

Date: July 21, 2014
     -----------------

* Print the name and title of each signing officer under his or her signature.