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, 2012
                                              --------------


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

                                                                     SEMI-ANNUAL
                                                                          REPORT
                                                              For the Six Months
                                                              Ended May 31, 2012


                                                                     FIRST TRUST

                                                     Energy Income Partners, LLC



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

                  FIRST TRUST ENERGY INFRASTRUCTURE FUND (FIF)
                               SEMI-ANNUAL REPORT
                                  MAY 31, 2012

Shareholder Letter...........................................................  1
At A Glance..................................................................  2
Portfolio Commentary.........................................................  3
Portfolio of Investments.....................................................  6
Statement of Assets and Liabilities.......................................... 10
Statement of Operations...................................................... 11
Statement 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 Notes to
Financial Statements 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, 2012


Dear Shareholders:

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

The report you hold contains detailed information about your investment; a
portfolio commentary from the Fund's management team that provides a recap of
the period; a performance analysis and a market and Fund outlook. Additionally,
you will find the Fund's financial statements for the period this report covers.
I encourage you to read this document and discuss it with your financial
advisor. A successful investor is also typically a knowledgeable one, as we have
found to be the case at First Trust.

First Trust remains committed to being a long-term investor and investment
manager and to bringing you quality financial solutions regardless of market ups
and downs. We have always believed, as I have written previously, that there are
two ways to attain success in reaching your financial goals: staying invested in
quality products and having a long-term investment horizon. We are committed to
this approach in the products we manage or supervise and offer to investors.

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

First Trust will continue to make up-to-date information about your investments
available 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. I look forward to
the remainder of 2012 and to the next edition of your Fund's report.

Sincerely,

/s/ James A. Bowen

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

                                                                          Page 1






FIRST TRUST ENERGY INFRASTRUCTURE FUND
"AT A GLANCE"
AS OF MAY 31, 2012 (UNAUDITED)

------------------------------------------------------------------
FUND STATISTICS
------------------------------------------------------------------
Symbol on New York Stock Exchange                              FIF
Common Share Price                                          $20.35
Common Share Net Asset Value ("NAV")                        $21.88
Premium (Discount) to NAV                                    (6.99)%
Net Assets Applicable to Common Shares                $384,038,985
Current Monthly Distribution per Common Share (1)          $0.1085
Current Annualized Distribution per Common Share           $1.3020
Current Distribution Rate on Closing Common Share Price (2)   6.40%
Current Distribution Rate on NAV (2)                          5.95%
------------------------------------------------------------------

------------------------------------------------------------------
COMMON SHARE PRICE & NAV (WEEKLY CLOSING PRICE)
------------------------------------------------------------------
         Common Share Price       NAV
11/11    19.64                    20.43
         19.84                    21.35
         20.00                    21.37
         20.05                    21.34
         20.35                    22.35
12/11    20.25                    22.47
         21.08                    22.31
         20.83                    21.95
         20.78                    22.19
1/12     21.08                    22.22
         21.15                    22.39
         21.36                    22.45
         21.49                    22.76
2/12     21.74                    22.94
         21.96                    22.92
         21.63                    22.93
         21.80                    22.57
         21.16                    22.56
3/12     21.29                    22.52
         21.25                    22.55
         20.75                    22.10
         20.97                    22.55
4/12     21.63                    23.03
         21.18                    22.67
         21.15                    22.60
         19.94                    21.66
         20.26                    22.16
5/12     20.35                    21.88


------------------------------------------------------------------
PERFORMANCE
------------------------------------------------------------------
                                     Cumulative Total Return
                                ----------------------------------
                                                       Inception
                                6 Months Ended        (9/27/2011)
                                  5/31/2012           to 5/31/2012
Fund Performance (3)
NAV                                 5.55%                18.15%
Market Value                        5.90%                4.95%

Index Performance
Philadelphia Stock Exchange
   Utility Index                    4.26%                8.06%
Alerian MLP Total Return Index      2.05%                9.67%
Blended Benchmark (4)               3.16%                8.87%
------------------------------------------------------------------


----------------------------------------------------------
                                                % OF TOTAL
INDUSTRY CLASSIFICATION                        INVESTMENTS
----------------------------------------------------------
Pipelines                                           60.6%
Electric Power                                      29.9
Propane                                              4.0
Marine                                               2.2
Coal                                                 1.8
Other                                                1.5
----------------------------------------------------------
                                      Total        100.0%
                                                   ======


----------------------------------------------------------
                                                % OF TOTAL
TOP 10 HOLDINGS                                INVESTMENTS
----------------------------------------------------------
Kinder Morgan Management, LLC                        7.5%
Enbridge Energy Management, LLC                      7.1
Williams Cos., Inc.                                  4.4
TransCanada Corp.                                    4.0
Dominion Resources, Inc.                             4.0
NextEra Energy, Inc.                                 3.9
NiSource, Inc.                                       3.5
Centerpoint Energy, Inc.                             3.2
Spectra Energy Corp.                                 3.1
Southern Co.                                         3.1
----------------------------------------------------------
                                     Total         43.8%
                                                   =====



(1)   Most recent distribution paid or declared through 5/31/2012. 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 5/31/2012. 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, 2012

                                  SUB-ADVISOR


ENERGY INCOME PARTNERS, LLC

Energy Income Partners, LLC ("EIP"), Westport, CT, was founded in 2003 to
provide professional asset management services in the area of energy-related
master limited partnerships and limited liability companies taxed as
partnerships ("MLPs") and other high-payout securities such as pipeline
companies, utilities and Canadian infrastructure equities. EIP mainly focuses on
investments in energy-related infrastructure assets such as pipelines, petroleum
storage and terminals that receive fee-based or regulated income from their
corporate customers. EIP managed or supervised approximately $1.6 billion of
assets, as of May 31, 2012. The other funds advised by EIP include a partnership
for U.S. high net worth individuals and a master-and-feeder fund for
institutions. EIP also manages separately managed accounts. EIP is a registered
investment advisor and serves as an advisor or sub-advisor to two registered
investment companies other than the Fund.

                           PORTFOLIO MANAGEMENT TEAM

JAMES J. MURCHIE
FOUNDER AND CEO OF ENERGY INCOME PARTNERS, LLC

Mr. Murchie founded EIP in 2003 and is the portfolio manager for all funds
advised by EIP which focus on energy-related master limited partnerships and
other high payout securities of companies that operate energy infrastructure.
From 2005 to mid-2006, Mr. Murchie and the EIP investment team joined Pequot
Capital Management. In July 2006, Mr. Murchie and the EIP investment team left
Pequot and re-established EIP. From 1998 to 2003, Mr. Murchie managed a
long/short fund that invested in energy and cyclical equities and commodities.
From 1995 to 1997, he was a managing director at Tiger Management where his
primary responsibilities were investments in energy, commodities and related
equities. From 1990 to 1995, Mr. Murchie was a principal at Sanford C. Bernstein
where he was a top-ranked energy analyst and sat on the Research Department's
Recommendation Review Committee. Before joining Bernstein, he spent 8 years at
British Petroleum in 7 operating and staff positions of increasing
responsibility. He has served on the board of Clark Refining and Marketing
Company and as President and Treasurer of the Oil Analysts Group of New York.
Mr. Murchie holds degrees from Rice University and Harvard University.

EVA PAO
PRINCIPAL OF ENERGY INCOME PARTNERS, LLC

Ms. Pao has been with EIP since its inception in 2003 and is co-portfolio
manager for all of the funds advised by EIP. She joined EIP in 2003, serving as
Managing Director of EIP until the EIP investment team joined Pequot Capital
Management. From 2005 to mid-2006, Ms. Pao served as Vice President of Pequot
Capital Management. Prior to Harvard Business School, Ms. Pao was a Manager at
Enron Corp where she managed a portfolio in Canadian oil and gas equities for
Enron's internal hedge fund that specialized in energy-related equities and
managed a natural gas trading book. She received a B.A. from Rice University in
1996. She received an M.B.A. from the Harvard Business School in 2002.

                                   COMMENTARY

FIRST TRUST ENERGY INFRASTRUCTURE FUND

The investment objective of First Trust Energy Infrastructure Fund ("FIF" or 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 MLPs, MLP
affiliates, Canadian income trusts and their successor companies, 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 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"), the total return for energy-related MLPs
and utilities over the semi-annual period ended May 31, 2012 was 2.05% and
4.26%, respectively. These figures are according to data collected from Alerian
Capital Management and Bloomberg. While in the short term, share appreciation

                                                                          Page 3






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

can be volatile, we believe that over the longer term, share appreciation will
approximate growth in per share quarterly cash distributions. Over the last 10
years, growth in per share MLP distributions and utility dividends has averaged
about 6.8% and 6.1%, respectively. Over the last 12 months, the cash
distributions of MLPs and utilities increased by about 5.6% and 4.9%,
respectively (source: Alerian Capital Management and Bloomberg).

PERFORMANCE ANALYSIS

On a net asset value ("NAV") basis, the Fund provided a total return1 of 5.55%,
including the reinvestment of dividends, for the semi-annual period ended May
31, 2012. This compares, according to collected data, to a total return of 3.16%
for the average of the two benchmarks (2.05% for AMZX and 4.26% for UTY). On a
market value basis, the Fund had a total return1, including the reinvestment of
dividends, for the period covered by this report, of 5.90%. The Fund's discount
to NAV narrowed slightly over the course of the semi-annual period. On May 31,
2012, the Fund was priced at $20.35 while the NAV was $21.88, a discount of
6.99%. As a comparison, on November 30, 2011, the Fund was priced at $19.82,
while the NAV was $21.38, a discount of 7.30%.

The Fund declared regular monthly common share distributions of $0.1085 per
share for each month of the reporting period.

The outperformance of the Fund's NAV relative to the 3.16% average of the AMZX
and UTY benchmarks is largely driven by the non-MLPs in the portfolio, as well
as ownership of companies that primarily have non-cyclical and/or regulated
businesses like pipelines and storage terminals, while underweighting or
avoiding altogether certain MLPs that we view as less attractive investments
(including some that operate in some of the non-cyclical businesses, but whose
individual company characteristics are not attractive.) We believe the most
important long-term drivers of the Fund's total return will be the maintenance
and growth of the dividends of the Fund's portfolio companies.

An important factor affecting the return of the Fund is the Fund's use of
financial leverage through the use of a line of credit. The Fund entered into a
committed facility agreement with the Bank of Nova Scotia that has a maximum
commitment amount of $145,000,000. The Fund uses leverage because its managers
believe 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 the share price. For example, as the prices of securities
held by the Fund decline, the effect of changes in Common Share NAV and Common
shareholder total return 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

The MLP asset class has experienced 5 IPOs in the six month reporting period.
There also has been a healthy level of secondary financing activity for MLPs
during the reporting period, as they continue to fund their ongoing investments
in new pipelines, processing and storage facilities. In the semi-annual period,
there have been 27 secondary equity offerings for MLPs that raised $8.0 billion
through May 31, 2012. This compares to $8.7 billion raised in 34 secondary
equity offerings during the same time period last year. MLPs also found access
to the public debt markets, raising $7.9 billion in 9 offerings during the same
time period. This compares to $9.0 billion during the same period last year
(source: Barclays Capital). The combination of equity and debt raised of
approximately $16 billion represents about 6% of the roughly $274 billion MLP
market cap. Since MLPs pay out virtually all of their free cash flow each
quarter, this capital raising is mostly a reflection of the capital investment
and acquisition activity in the asset class.

Capital spending for utilities continues to increase. As measured by the UTY,
capital expenditures have grown from $34 billion and $45 billion in 2002 and
2007, respectively, to $60 billion last year. This growth in expenditures is 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 made in their rate
structure.

The Fund continues to aim to be invested in 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

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 net asset
      value returns and changes in Common Share price for market value returns.
      Total returns do not reflect sales load. Past performance is not
      indicative of future results.

Page 4




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

unpredictable cyclical cash flows make them a poor fit, in the opinion of the
Sub-Advisor. While there are some businesses within the Fund's portfolio whose
cash flows have cyclicality, they are usually small and analyzed in the context
of each company's financial and operating leverage and payout ratio.

MLPs continue to play an integral role in the restructuring of some more
diversified energy conglomerates. The first phase of this restructuring, which
has occurred over the last few years, has been the creation by diversified
conglomerates of an MLP subsidiary that contains assets such as pipelines and
storage terminals. The more recent phase has been the divestiture by some of the
conglomerates 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. The diversified energy conglomerates are
restructuring so that their regulated infrastructure assets with predictable
cash flows may be better valued by the market. And sometimes, a more active role
is being taken as three diversified energy companies have been purchased by MLP
general partners over the last eight months. We expect this restructuring will
continue. The Fund is pursuing these opportunities in the context of the desired
characteristics including business exposure and quality.

The total return proposition of owning Energy Infrastructure Companies has been
and continues to be their yield plus their growth. The yield of the MLPs,
weighted by market capitalization, on May 31, 2012, was 6.6% based on AMZX and
for utilities, as measured by UTY, was 4.21%. The growth in the quarterly cash
distributions that make up this yield has averaged 6.8% annually over the last
ten years for MLPs and 6.1% for utilities. In the opinion of the Sub-Advisor,
this growth has been and will continue to be driven by three factors: 1) modest
increases in volume from growth from both underlying petroleum demand as the
economy recovers and increases in North American oil and gas production
(particularly in new supply areas); 2) inflation and cost escalators in pipeline
tariffs and contracts; and 3) accretion from profitable capital projects and
acquisitions. The capital projects continue to be related to growth in areas
such as the Canadian Oil Sands, the new oil and natural gas shale resources and
the need for more infrastructure related to bio fuels and other environmental
mandates, including the conversion of many coal fired power plants to natural
gas.

                                                                          Page 5





FIRST TRUST ENERGY INFRASTRUCTURE FUND
PORTFOLIO OF INVESTMENTS
MAY 31, 2012 (UNAUDITED)



   SHARES/
    UNITS                                       DESCRIPTION                                          VALUE
------------    ---------------------------------------------------------------------------     --------------
COMMON STOCKS - 100.6%

                ELECTRIC UTILITIES - 24.4%
                                                                                          
     651,800    Duke Energy Corp. (a)......................................................     $   14,326,564
     284,600    Emera, Inc. (CAD) (a)......................................................          9,106,870
     296,450    Exelon Corp. (a)...........................................................         10,962,721
     109,800    Fortis, Inc. (CAD) (a).....................................................          3,497,526
     115,000    ITC Holdings Corp. (a).....................................................          7,929,250
     300,400    NextEra Energy, Inc. (a)...................................................         19,628,136
     355,800    Northeast Utilities (a)....................................................         12,812,358
     340,000    Southern Co. (a)...........................................................         15,609,400
                                                                                                --------------
                                                                                                    93,872,825
                                                                                                --------------

                GAS UTILITIES - 5.5%
      15,800    ONEOK, Inc. (a)............................................................          1,311,242
     285,000    Questar Corp. (a)..........................................................          5,719,950
     492,600    UGI Corp. (a)..............................................................         14,127,768
                                                                                                --------------
                                                                                                    21,158,960
                                                                                                --------------

                MULTI-UTILITIES - 19.7%
      35,000    Atco, Ltd. (CAD) (a).......................................................          2,419,180
      45,500    Canadian Utilities, Ltd. (CAD) (a).........................................          2,951,982
     797,600    Centerpoint Energy, Inc. (a)...............................................         16,135,448
     391,400    Dominion Resources, Inc. (a)...............................................         20,376,284
     195,000    National Grid PLC, ADR (a).................................................          9,806,550
     708,300    NiSource, Inc. (a).........................................................         17,771,247
      50,000    Sempra Energy (a)..........................................................          3,250,500
      75,000    Wisconsin Energy Corp. (a).................................................          2,838,000
                                                                                                --------------
                                                                                                    75,549,191
                                                                                                --------------

                OIL, GAS & CONSUMABLE FUELS - 51.0%
   1,157,163    Enbridge Energy Management, LLC (a) (b)....................................         36,091,891
     534,700    Enbridge Income Fund Holdings, Inc. (CAD) (a)..............................         12,419,473
     353,169    Enbridge, Inc. (a).........................................................         13,925,454
     252,300    Keyera Corp. (CAD) (a).....................................................         10,271,787
     534,593    Kinder Morgan Management, LLC (b)..........................................         37,972,144
     306,500    Kinder Morgan, Inc. (a)....................................................         10,479,235
     482,100    Pembina Pipeline Corp. (CAD) (a)...........................................         12,990,118
     552,350    Spectra Energy Corp. (a)...................................................         15,857,969
     500,400    TransCanada Corp. (a)......................................................         20,476,368
     258,000    Veresen, Inc. (CAD) (a)....................................................          3,242,329
     730,700    Williams Cos., Inc. (a)....................................................         22,308,271
                                                                                                --------------
                                                                                                   196,035,039
                                                                                                --------------

                TOTAL COMMON STOCKS .......................................................        386,616,015
                (Cost $353,093,443)                                                             --------------

MASTER LIMITED PARTNERSHIPS - 31.5%

                GAS UTILITIES - 1.6%
     157,764    AmeriGas Partners, L.P. (a)................................................         6,081,8026
                                                                                                --------------



Page 6                 See Notes to Financial Statements




FIRST TRUST ENERGY INFRASTRUCTURE FUND
PORTFOLIO OF INVESTMENTS - (CONTINUED)
MAY 31, 2012 (UNAUDITED)



   SHARES/
    UNITS                                       DESCRIPTION                                          VALUE
------------    ---------------------------------------------------------------------------     --------------
MASTER LIMITED PARTNERSHIPS - (CONTINUED)

                OIL, GAS & CONSUMABLE FUELS - 29.9%
                                                                                          
      20,000    Alliance GP Holdings, L.P. ................................................     $      831,400
      72,165    Alliance Resource Partners, L.P.(a) .......................................          4,114,126
     160,460    Buckeye Partners, L.P. (a).................................................          7,625,059
     162,000    El Paso Pipeline Partners, L.P. (a) .......................................          5,315,220
     149,000    Energy Transfer Equity, L.P. (a)...........................................          5,413,170
      51,600    Enterprise Products Partners, L.P. (a).....................................          2,516,016
     128,423    Holly Energy Partners, L.P. (a)............................................          7,264,889
      99,300    Magellan Midstream Partners, L.P. (a)......................................          6,832,833
     173,239    Natural Resource Partners, L.P. ...........................................          3,974,103
     213,200    NuStar Energy, L.P. (a)....................................................         11,131,172
      29,600    NuStar GP Holdings, LLC ...................................................            943,944
      16,000    ONEOK Partners, L.P. ......................................................            873,600
     144,443    Plains All American Pipeline, L.P. (a).....................................         11,343,109
     117,700    Spectra Energy Partners, L.P. (a)..........................................          3,668,709
     177,909    Sunoco Logistics Partners, L.P. (a)........................................          5,991,975
     353,095    TC Pipelines, L.P. (a).....................................................         14,476,895
     304,828    Teekay LNG Partners, L.P. (a)..............................................         11,373,133
     242,670    TransMontaigne Partners, L.P. (a)..........................................          7,653,812
      66,300    Williams Partners, L.P. (a)................................................          3,507,270
                                                                                                --------------
                                                                                                   114,850,435
                                                                                                --------------

                TOTAL MASTER LIMITED PARTNERSHIPS .........................................        120,932,237
                (Cost $116,926,667)                                                             --------------


                TOTAL INVESTMENTS - 132.1% ................................................        507,548,252
                (Cost $470,020,110) (c)                                                         --------------

  NUMBER OF
  CONTRACTS                                     DESCRIPTION                                          VALUE
------------    ---------------------------------------------------------------------------     --------------
 CALL OPTIONS WRITTEN - (0.3%)

                Centerpoint Energy, Inc. Call
         300    @ 22.5 due November 12 ....................................................             (9,900)
                                                                                                --------------

                Dominion Resources, Inc. Calls
         400    @ 52.5 due June 12 ........................................................            (10,000)
       1,853    @ 55 due July 12 ..........................................................            (18,530)
                                                                                                --------------
                                                                                                       (28,530)
                                                                                                --------------
                Duke Energy Corp. Calls
       2,000    @ 22 due July 12 ..........................................................            (90,000)
         400    @ 23 due July 12 ..........................................................             (3,000)
         800    @ 23 due October 12 .......................................................            (24,000)
                                                                                                --------------
                                                                                                      (117,000)
                                                                                                --------------
                Enbridge, Inc. Call
       1,300    @ 45 due October 12 .......................................................            (26,000)
                                                                                                --------------

                Exelon Corp. Calls
         600    @ 38 due July 12 ..........................................................            (27,000)
         183    @ 42 due July 12 ..........................................................               (915)
         210    @ 42 due October 12 .......................................................             (2,100)
                                                                                                --------------
                                                                                                       (30,015)
                                                                                                --------------



                       See Notes to Financial Statements                  Page 7




FIRST TRUST ENERGY INFRASTRUCTURE FUND
PORTFOLIO OF INVESTMENTS - (CONTINUED)
MAY 31, 2012 (UNAUDITED)



  NUMBER OF
  CONTRACTS                                   DESCRIPTION                                            VALUE
------------  -----------------------------------------------------------------------------     --------------

 CALL OPTIONS WRITTEN - (CONTINUED)

                Kinder Morgan, Inc. Calls
                                                                                          
          42    @ 35 due September 12 .....................................................     $       (5,670)
         800    @ 40 due September 12 .....................................................            (24,000)
         400    @ 35 due December 12 ......................................................            (58,000)
                                                                                                --------------
                                                                                                       (87,670)
                                                                                                --------------
                National Grid PLC, ADR Call
       1,050    @ 55 due September 12 .....................................................            (49,350)
                                                                                                --------------

                NextEra Energy, Inc. Calls
       1,336    @ 65 due June 12 ..........................................................           (146,960)
         490    @ 65 due September 12 .....................................................           (120,050)
                                                                                                --------------
                                                                                                      (267,010)
                                                                                                --------------
                NiSource, Inc. Call
         883    @ 25 due July 12 ..........................................................            (79,470)
                                                                                                --------------

                Questar Corp. Calls
       1,564    @ 21 due July 12 ..........................................................            (31,280)
         100    @ 21 due October 12 .......................................................             (5,500)
                                                                                                --------------
                                                                                                       (36,780)
                                                                                                --------------
                Southern Co. Calls
         800    @ 46 due July 12 ..........................................................            (72,000)
         655    @ 47 due August 12 ........................................................            (34,060)
                                                                                                --------------
                                                                                                      (106,060)
                                                                                                --------------
                Spectra Energy Corp. Calls
       2,500    @ 33 due June 12 ..........................................................            (25,000)
         200    @ 33 due September 12 .....................................................             (5,000)
         400    @ 31 due December 12 ......................................................            (38,000)
                                                                                                --------------
                                                                                                       (68,000)
                                                                                                --------------
                TransCanada Corp. Call
       1,300    @ 45 due August 12 ........................................................            (19,500)
                                                                                                --------------

                UGI Corp. Calls
         200    @ 30 due June 12 ..........................................................             (2,000)
       1,352    @ 30 due July 12 ..........................................................            (23,660)
       1,375    @ 30 due October 10 .......................................................            (82,500)
                                                                                                --------------
                                                                                                      (108,160)
                                                                                                --------------
                Williams Cos., Inc. Calls
       1,000    @ 35 due June 12 ..........................................................             (3,000)
       1,000    @ 34 due August 12 ........................................................            (45,000)
         900    @ 34 due November 12 ......................................................            (90,000)
                                                                                                --------------
                                                                                                      (138,000)
                                                                                                --------------

                TOTAL CALL OPTIONS WRITTEN ................................................         (1,171,445)
                (Premiums received $1,061,285)                                                  --------------

                OUTSTANDING LOAN - (32.0%) ................................................       (123,000,000)
                NET OTHER ASSETS AND LIABILITIES - 0.2% ...................................            662,178
                                                                                                --------------
                NET ASSETS - 100.0% .......................................................     $  384,038,985
                                                                                                --------------



Page 8           See Notes to Financial Statements




FIRST TRUST ENERGY INFRASTRUCTURE FUND
PORTFOLIO OF INVESTMENTS - (CONTINUED)
MAY 31, 2012 (UNAUDITED)


(a)   All or a portion of this security is available to serve as collateral on
      the outstanding loan.

(b)   Non-income producing security which pays in-kind distributions.

(c)   Aggregate cost for financial reporting purposes, which approximates the
      aggregate cost for federal income tax purposes. As of May 31, 2012, the
      aggregate gross unrealized appreciation for all securities in which there
      was an excess of value over tax cost was $45,269,264 and the aggregate
      gross unrealized depreciation for all securities in which there was an
      excess of tax cost over value was $7,741,122.

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.

VALUATION INPUTS

A summary of the inputs used to value the Fund's investments as of May 31, 2012
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
                                                          5/31/2012          PRICES            INPUTS            INPUTS
                                                       --------------    --------------    --------------    --------------
Common Stocks*....................................     $  386,616,015    $  386,616,015    $           --    $           --
Master Limited Partnerships*......................        120,932,237       120,932,237                --                --
                                                       --------------    --------------    --------------    --------------
TOTAL INVESTMENTS..............................        $  507,548,252    $  507,548,252    $           --    $           --
                                                       ==============    ==============    ==============    ==============

                                                     LIABILITIES TABLE
                                                                                            LEVEL 2            LEVEL 3
                                                            TOTAL          LEVEL 1         SIGNIFICANT        SIGNIFICANT
                                                          VALUE AT          QUOTED         OBSERVABLE        UNOBSERVABLE
                                                          5/31/2012         PRICES           INPUTS             INPUTS
                                                       --------------    --------------    --------------    --------------
Call Options Written...........................        $   (1,171,445)   $   (1,171,445)   $           --    $           --
                                                       ==============    ==============    ==============    ==============


* See Portfolio of Investments for industry breakout.

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. As of May 31,
2012, the Fund transferred common stocks valued at $27,713,087 from Level 2 to
Level 1 of the fair value hierarchy. The common stock that transferred from
Level 2 to Level 1 did so as a result of the availability of quoted prices in an
active market. This common stock was previously fair valued in accordance with
procedures adopted by the Fund's Board of Trustees and the provisions of the
Investment Company Act of 1940, as amended.

                       See Notes to Financial Statements                  Page 9




FIRST TRUST ENERGY INFRASTRUCTURE FUND
STATEMENT OF ASSETS AND LIABILITIES
MAY 31, 2012 (UNAUDITED)



ASSETS:
                                                                                                
Investments, at value
  (Cost $470,020,110) .........................................................................    $ 507,548,252
Cash ..........................................................................................        7,033,509
Prepaid expenses ..............................................................................           20,837
Receivables:
    Dividends..................................................................................        1,997,970
    Investment securities sold.................................................................        1,041,777
    Interest...................................................................................              762
                                                                                                   -------------
       Total Assets............................................................................      517,643,107
                                                                                                   -------------

LIABILITIES:
Outstanding loan ..............................................................................      123,000,000
Options written, at value (Premiums received $1,061,285) ......................................        1,171,445
Payables:
    Investment securities purchased............................................................        8,615,068
    Investment advisory fees...................................................................          436,768
    Offering costs.............................................................................          103,795
    Interest and fees on loan..................................................................           97,894
    Custodian fees.............................................................................           86,612
    Administrative fees........................................................................           35,312
    Audit and tax fees.........................................................................           30,654
    Legal fees.................................................................................            8,781
    Printing fees..............................................................................            7,913
    Trustees' fees and expenses................................................................            5,960
    Transfer agent fees........................................................................            1,848
    Financial reporting fees...................................................................              771
Other liabilities                                                                                          1,301
                                                                                                   -------------
      Total Liabilities........................................................................      133,604,122
                                                                                                   -------------
NET ASSETS.....................................................................................    $ 384,038,985
                                                                                                   =============

NET ASSETS CONSIST OF:
Paid-in capital ...............................................................................    $ 334,344,006
Par value .....................................................................................          175,502
Accumulated net investment income (loss) ......................................................       (8,481,960)
Accumulated net realized gain (loss) on investments, written options and foreign
  currency transactions .......................................................................       20,588,682
Net unrealized appreciation (depreciation) on investments, written options and
 foreign currency translation .................................................................       37,412,755
                                                                                                   -------------
NET ASSETS ....................................................................................    $ 384,038,985
                                                                                                   =============
NET ASSET VALUE, per Common Share (par value $0.01 per Common Share) ..........................    $       21.88
                                                                                                   =============
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
STATEMENT OF OPERATIONS
FOR THE SIX MONTHS ENDED MAY 31, 2012 (UNAUDITED)


INVESTMENT INCOME:
                                                                                                
Dividends (net of foreign withholding tax of $292,379).........................................    $   5,990,160
Interest.......................................................................................            4,182
Other..........................................................................................               20
                                                                                                   -------------
   Total investment income.....................................................................        5,994,362
                                                                                                   -------------

EXPENSES:
Investment advisory fees.......................................................................        2,548,892
Interest and fees on loan......................................................................          571,060
Administrative fees............................................................................          206,167
Custodian fees.................................................................................           57,252
Legal fees.....................................................................................           49,743
Printing fees..................................................................................           42,104
Audit and tax fees.............................................................................           24,818
Trustees' fees and expenses....................................................................           18,324
Transfer agent fees............................................................................           14,413
Financial reporting fees.......................................................................            4,625
Other..........................................................................................           23,083
                                                                                                   -------------
   Total expenses..............................................................................        3,560,481
                                                                                                   -------------
NET INVESTMENT INCOME (LOSS)...................................................................        2,433,881
                                                                                                   -------------
NET REALIZED AND UNREALIZED GAIN (LOSS):
Net realized gain (loss) on:
   Investments.................................................................................       14,099,335
   Written options (a).........................................................................        1,466,079
   Foreign currency transactions...............................................................          (86,561)
                                                                                                   -------------
Net realized gain (loss).......................................................................       15,478,853
                                                                                                   -------------
Net change in unrealized appreciation (depreciation) on:
   Investments.................................................................................        2,155,811
   Written options (a).........................................................................          170,370
   Foreign currency translation................................................................           (6,925)
                                                                                                   -------------
Net change in unrealized appreciation (depreciation)...........................................        2,319,256
                                                                                                   -------------
NET REALIZED AND UNREALIZED GAIN (LOSS)........................................................       17,798,109
                                                                                                   -------------
NET INCREASE (DECREASE) IN NET ASSETS RESULTING FROM OPERATIONS................................    $  20,231,990
                                                                                                   =============


(a)   Primary risk exposure is equity option contracts.


                       See Notes to Financial Statements                 Page 11




FIRST TRUST ENERGY INFRASTRUCTURE FUND
STATEMENT OF CHANGES IN NET ASSETS



                                                                                          SIX MONTHS
                                                                                             ENDED          PERIOD
                                                                                           5/31/2012         ENDED
                                                                                          (UNAUDITED)   11/30/2011 (a)
                                                                                        --------------  --------------
OPERATIONS:
                                                                                                  
Net investment income (loss).......................................................     $    2,433,881  $      912,374
Net realized gain (loss)...........................................................         15,478,853      4,706,818
Net change in unrealized appreciation (depreciation)...............................          2,319,256     35,093,499
                                                                                        --------------  --------------
Net increase (decrease) in net assets resulting from operations....................         20,231,990     40,712,691
                                                                                        --------------  --------------

DISTRIBUTIONS TO SHAREHOLDERS FROM:
Net investment income..............................................................        (11,425,204)             --
Net realized gain..................................................................                 --              --
Return of capital..................................................................                 --              --
                                                                                        --------------  --------------
Total distributions to shareholders................................................        (11,425,204)             --
                                                                                        --------------  --------------

CAPITAL TRANSACTIONS:
Proceeds from Common Shares sold ..................................................                 --     335,209,508 (b)
Proceeds from Common Shares reinvested.............................................                 --              --
Offering costs.....................................................................                 --        (690,000)
                                                                                        --------------  --------------
Net increase (decrease) in net assets resulting from capital transactions..........                 --     334,519,508
                                                                                        --------------  --------------
Total increase (decrease) in net assets............................................          8,806,786     375,232,199

NET ASSETS:
Beginning of period................................................................        375,232,199              --
                                                                                        --------------  --------------
End of period......................................................................     $  384,038,985  $  375,232,199
                                                                                        ==============  ==============
Accumulated net investment income (loss) at end of period..........................     $   (8,481,960) $      509,363
                                                                                        ==============  ==============

CAPITAL TRANSACTIONS WERE AS FOLLOWS:
Common Shares at beginning of period...............................................         17,550,236              --
Common Shares sold.................................................................                 --      17,550,236 (b)
Common Shares issued as reinvestment under the Dividend Reinvestment Plan..........                 --              --
                                                                                        --------------  --------------
Common Shares at end of period.....................................................         17,550,236      17,550,236
                                                                                        ==============  ==============




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

(a)   The Fund was seeded on August 18, 2011 and commenced operations on
      September 27, 2011.

(b)   Includes 295,000 shares sold from the over allotment option of the initial
      public offering. The shares were sold on November 10, 2011, the trade
      date, at the initial offering price of $19.10, which differed from the
      closing common share price of $20.01 and the closing NAV per share of
      $20.85 on that date.

Page 12                 See Notes to Financial Statements




FIRST TRUST ENERGY INFRASTRUCTURE FUND
STATEMENT OF CASH FLOWS
FOR THE SIX MONTHS ENDED MAY 31, 2012 (UNAUDITED)




CASH FLOWS FROM OPERATING ACTIVITIES:
                                                                                                  
Net increase (decrease) in net assets resulting from operations....................     $   20,231,990
Adjustments to reconcile net increase (decrease) in net assets resulting from
 operations to net cash used in operating activities:
   Purchases of investments........................................................       (126,222,124)
   Sales of investments............................................................         86,834,127
   Proceeds from written options...................................................          2,196,548
   Cost of options closed..........................................................           (622,183)
   Return of capital received from investment in MLPs..............................          3,669,920
   Net realized gain/loss on investments and options...............................        (15,565,414)
   Net change in unrealized appreciation/depreciation on investments
     and options...................................................................         (2,326,181)

CHANGES IN ASSETS AND LIABILITIES:
   Decrease in interest receivable.................................................              5,992
   Increase in dividends receivable (a)............................................           (731,186)
   Increase in prepaid expenses....................................................            (14,738)
   Increase in interest and fees on loan payable...................................             27,874
   Increase in investment advisory fees payable....................................             74,653
   Decrease in audit and tax fees payable..........................................            (19,846)
   Increase in legal fees payable..................................................              2,713
   Decrease in printing fees payable...............................................            (16,207)
   Increase in administrative fees payable.........................................              5,699
   Increase in custodian fees payable..............................................             57,252
   Decrease in offering cost payable...............................................           (421,709)
   Decrease in transfer agent fees payable.........................................               (319)
   Decrease in Trustees' fees and expenses payable.................................               (470)
   Decrease in financial reporting fees payable....................................               (719)
   Increase in other liabilities payable...........................................                457
                                                                                        --------------
CASH USED IN OPERATING ACTIVITIES..................................................                     $  (32,833,871)
                                                                                                        --------------
CASH FLOWS FROM FINANCING ACTIVITIES:
   Distributions to Common Shareholders from net investment income.................        (11,425,204)
   Issuances of loan...............................................................         21,000,000
                                                                                        --------------
CASH FLOWS PROVIDED BY FINANCING ACTIVITIES........................................                          9,574,796
                                                                                                        --------------
Decrease in cash...................................................................                        (23,259,075)
Cash at beginning of period........................................................                         30,292,584
                                                                                                        --------------
CASH AT END OF PERIOD..............................................................                     $    7,033,509
                                                                                                        ==============
SUPPLEMENTAL DISCLOSURE OF CASH FLOW INFORMATION:
Cash paid during the period for interest and fees..................................                     $      543,186
                                                                                                        ==============




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

(a)   Includes net change in unrealized appreciation (depreciation) on foreign
      currency of ($6,925).


                       See Notes to Financial Statements                 Page 13



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


                                                                         SIX MONTHS
                                                                            ENDED          PERIOD
                                                                          5/31/2012         ENDED
                                                                         (UNAUDITED)    11/30/2011 (a)
                                                                        -------------   -------------
                                                                                    
Net asset value, beginning of period ...                                  $   21.38       $   19.10 (b)
                                                                          ---------       ---------
INCOME FROM INVESTMENT OPERATIONS:
Net investment income (loss) .......................................           0.14            0.05
Net realized and unrealized gain (loss) ............................           1.01            2.30
                                                                          ---------       ---------
Total from investment operations ...................................           1.15            2.35
                                                                          ---------       ---------
Common Shares offering costs charged to paid-in capital.............             --           (0.04)
                                                                          ---------       ---------
Capital reduction resulting from issuance of Common Shares
    related to over allotment ......................................             --           (0.03)
                                                                          ---------       ---------
DISTRIBUTIONS PAID TO SHAREHOLDERS FROM:
Net investment income ..............................................          (0.65)             --
Net realized gain ..................................................             --              --
Return of capital ..................................................             --              --
                                                                          ---------       ---------
Total distributions.................................................          (0.65)             --
                                                                          ---------       ---------
Net asset value, end of period .....................................      $   21.88       $   21.38
                                                                          =========       =========
Market value, end of period ........................................      $   20.35       $   19.82
                                                                          =========       =========
TOTAL RETURN BASED ON NET ASSET VALUE (c)...........................           5.55%          11.94%
                                                                          =========       =========
TOTAL RETURN BASED ON MARKET VALUE (c)..............................           5.90%          (0.90)%
                                                                          =========       =========

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

RATIOS TO AVERAGE NET ASSETS/SUPPLEMENTAL DATA:
Net assets, end of period (in 000's) ...............................      $ 384,039       $ 375,232
Ratio of total expenses to average net assets ......................           1.82% (d)       1.66% (d)
Ratio of total expenses to average net assets excluding
   interest expense and fees on loan ...............................           1.53% (d)       1.48% (d)
Ratio of net investment income (loss) to average net assets ........           1.24% (d)       1.49% (d)
Portfolio turnover rate ............................................             18%              8%
INDEBTEDNESS:
Total loan outstanding (in 000's) ..................................      $ 123,000       $ 102,000
Asset coverage per $1,000 of indebtedness (e).......................      $   4,122       $   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)  Annualized.
(e)  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
                            MAY 31, 2012 (UNAUDITED)


                              1. FUND DESCRIPTION

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 ("SEC") 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 trusts
and their successor companies (collectively, "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 invests, 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. Domestic debt securities
and 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 in accordance with valuation procedures
adopted by the Fund's Board of Trustees, and in accordance with provisions of
the 1940 Act. The following securities, for which accurate and reliable market
quotations are readily available, will be valued as follows:

      Common stocks, MLPs and other securities listed on any national or foreign
      exchange (excluding the NASDAQ National Market ("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. If
      there are no transactions on the valuation day, the securities are valued
      at the mean between the most recent bid and asked prices.

      Securities listed on the NASDAQ or the AIM are valued at the official
      closing price. If there is no official closing price on the valuation day,
      the securities are valued at the mean between the most recent bid and
      asked prices.

      Securities traded in the over-the-counter market are valued at their
      closing bid prices.

      Exchange-traded options and futures 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 and futures contracts
      are valued at the mean between the most recent bid and asked prices.
      Over-the-counter options and futures contracts are valued at their closing
      bid prices.

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

All market quotations used in valuing the Fund's securities will be obtained
from a third party pricing service. If no quotation is received from a pricing
service, attempts will be made to obtain one or more broker quotes for the
security. In the event the pricing service does not provide a valuation, broker
quotations are not readily available, or the valuations received are deemed
unreliable, the Fund's Board of Trustees has designated First Trust Advisors
L.P. ("First Trust") to use a fair value method to value the Fund's securities.
Additionally, if events occur after the close of the principal markets for
certain securities (e.g., domestic debt and foreign securities) that could
materially affect the Fund's NAV, First Trust will use a fair value method to
value the Fund's securities. The use of fair value pricing is governed by
valuation procedures adopted by the Fund's Board of Trustees, and in accordance
with the provisions of the 1940 Act. As a general principle, the fair value of a
security is the amount which the Fund might reasonably expect to receive for the
security upon its current sale. However, in light of the judgment involved in
fair valuations, there can be no assurance that a fair value assigned to a
particular security will be the amount which the Fund might be able to receive
upon its current sale. Fair valuation of a security will be based on the
consideration of all available information, 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;


                                                                         Page 15



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

                     FIRST TRUST ENERGY INFRASTRUCTURE FUND
                            MAY 31, 2012 (UNAUDITED)

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, 2012, 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 enter into options written to hedge against changes
in the value of equities. Also, by writing (selling) options, the Fund seeks to
generate additional income, in the form of premiums received for writing
(selling) the options. The Fund may write (sell) covered call or put options
("options") on all or a portion of the common stock of energy companies 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 of energy companies 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 equity
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 equity 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 gain or loss. If the price
of the underlying equity 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

Page 16



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                     FIRST TRUST ENERGY INFRASTRUCTURE FUND
                            MAY 31, 2012 (UNAUDITED)

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 ("OTC") 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. 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.

D. RESTRICTED SECURITIES:

The Fund may invest up to 15% of its Managed Assets in restricted securities.
Managed Assets means the total asset value of the Fund minus the sum of the
Fund's liabilities other than the principal amount of borrowings. Restricted
securities are securities that may not be offered for public sale without first
being registered under the Securities Act of 1933, as amended (the "1933 Act").
Prior to registration, restricted securities may only be resold in transactions
exempt from registration under Rule 144A under the 1933 Act, normally to
qualified institutional buyers. As of May 31, 2012, the Fund held no restricted
securities.

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 and dividends
in connection with leverage, if any. 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.

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.

As of November 30, 2011, the components of distributable earnings and net assets
on a tax basis were as follows:

Undistributed ordinary income.....................  $   5,507,810
Undistributed capital gains.......................             --
                                                    -------------
Total undistributed earnings......................      5,507,810
Accumulated capital and other losses..............        (54,615)
Net unrealized appreciation (depreciation)........     35,259,496
                                                    -------------
Total accumulated earnings (losses)...............     40,712,691
Other   ..........................................             --
Paid-in capital...................................    334,519,508
                                                    -------------
Net assets........................................  $ 375,232,199
                                                    =============

F. INCOME TAXES:

The Fund intends 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 98.2% of the Fund's taxable income exceeds the distributions
from such taxable income for the calendar year.

                                                                         Page 17



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

                     FIRST TRUST ENERGY INFRASTRUCTURE FUND
                            MAY 31, 2012 (UNAUDITED)

The Fund intends to utilize provisions of the federal income tax laws which
allow it to carry realized capital losses forward indefinitely following the
year of the loss and offset such loss against any future realized capital gains.
At November 30, 2011, the Fund had no capital loss carryforward for federal
income tax purpose.

Certain capital losses realized after October 31 may be deferred and treated as
occurring on the first day of the following fiscal year. For the fiscal year
ended November 30, 2011, the Fund elected to defer net realized currency losses
of $54,615 incurred between November 1, 2011 and November 30, 2011.

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 year 2011 remains open to
federal and state audit. As of May 31, 2012 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 trade date and subsequent sale trade date is included in "Net
realized gain (loss) on foreign currency transactions" on the Statement of
Operations.

I. ORGANIZATION AND OFFERING COSTS:

Organization costs consisted of costs incurred to establish the Fund and enable
it to legally conduct business. These costs included filing fees, listing fees,
legal services pertaining to the organization of the business and audit fees
relating to the initial registration and auditing the initial statement of
assets and liabilities, among other fees. Offering costs consisted of legal fees
pertaining to the Fund's shares offered for sale, registration fees,
underwriting fees, and printing of the initial prospectus, among other fees.
First Trust paid all organization expenses. The Fund's Common Share offering
costs of $690,000 were recorded as a reduction of the proceeds from the sale of
Common Shares during the period ended November 30, 2011.

J. ACCOUNTING PRONOUNCEMENTS:

In May 2011, the Financial Accounting Standards Board ("FASB") issued ASU
2011-04, "Amendments to Achieve Common Fair Value Measurement and Disclosure
Requirements in U.S. GAAP and IFRSs," modifying Topic 820, "Fair Value
Measurements and Disclosures." At the same time, the International Accounting
Standards Board ("IASB") issued International Financial Reporting Standard
("IFRS") 13, "Fair Value Measurement." The objective of the FASB and IASB is
convergence of their guidance on fair value measurements and disclosures.
Specifically, the ASU requires reporting entities to disclose (i) the amounts of
any transfers between Level 1 and Level 2, and the reasons for the transfers,
(ii) for Level 3 fair value measurements, quantitative information about
significant unobservable inputs used, (iii) a description of the valuation
processes used by the reporting entity, and (iv) a narrative description of the
sensitivity of the fair value measurement to changes in unobservable inputs if a
change in those inputs might result in a significantly higher or lower fair
value measurement. The effective date of the ASU is for interim and annual
periods beginning after December 15, 2011, and it is therefore not effective for
the current fiscal year. Management is in the process of assessing the impact of
the updated standards on the Fund's financial statements, if any.

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

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. 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.
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 Funds Managed Assets that is paid
by First Trust out of its investment advisory fee.

Page 18



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

                     FIRST TRUST ENERGY INFRASTRUCTURE FUND
                            MAY 31, 2012 (UNAUDITED)

First Trust Capital Partners, LLC ("FTCP"), an affiliate of First Trust, owns,
through a wholly-owned subsidiary, a 15% ownership interest in each of the
Sub-Advisor and EIP Partners, LLC, an affiliate of the Sub-Advisor. In addition,
FTCP purchased a preferred interest in the Sub-Advisor. As of the date of this
report, the Sub-Advisor has bought back all of FTCP's preferred interest.

BNY Mellon Investment Servicing (US) Inc. serves as the Fund's Administrator,
Fund Accountant and Transfer Agent in accordance with certain fee arrangements.
The Bank of New York Mellon serves as the Fund's Custodian in accordance with
certain fee arrangements.

Effective January 23, 2012, James A. Bowen resigned from his position as the
President and Chief Executive Officer of the Fund. He will continue as a
Trustee, the Chairman of the Board of Trustees and a member of the Executive
Committee. The Board elected Mark R. Bradley to serve as the President and Chief
Executive Officer of the Fund and James M. Dykas to serve as the Treasurer,
Chief Financial Officer and Chief Accounting Officer of the Fund.

Effective January 1, 2012, 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 of $125,000 per year and an annual per fund fee
of $4,000 for each closed-end fund or other actively managed fund and $1,000 for
each index fund in the First Trust Fund Complex. The fixed annual retainer is
allocated pro rata among each fund in the First Trust Fund Complex based on net
assets. Prior to January 1, 2012, each Independent Trustee received an annual
retainer of $10,000 per trust for the first 14 trusts of the First Trust Fund
Complex and an annual retainer of $7,500 per trust for each additional trust in
the First Trust Fund Complex. The annual retainer was allocated equally among
each of the trusts.

Additionally, the Lead Independent Trustee is paid $15,000 annually, the
Chairman of the Audit Committee is paid $10,000 annually, and each of the
Chairmen of the Nominating and Governance Committee and the Valuation Committee
is paid $5,000 annually to serve in such capacities, with such compensation
allocated pro rata among each fund in the First Trust Fund Complex based on net
assets. Prior to January 1, 2012, the annual amounts paid were $10,000, $5,000
and $2,500, respectively. Trustees are reimbursed for travel and out-of-pocket
expenses in connection with all meetings. The Lead Independent Trustee and each
Committee chairman will serve two-year terms before rotating to serve as
chairman of another committee or as Lead Independent Trustee. The officers and
"Interested" Trustee receive no compensation from the funds for acting in such
capacities.

                      4. PURCHASES AND SALES OF SECURITIES

Cost of purchases and proceeds from sales of investment securities, excluding
short-term investments, for the six months ended May 31, 2012 were $130,642,493
and $87,820,005, respectively.

Written option activity for the Fund was as follows:


                                                     NUMBER
                                                       OF
WRITTEN OPTIONS                                     CONTRACTS       PREMIUMS
--------------------------------------------------------------------------------
Options outstanding at November 30, 2011...            29,526      $1,735,315
Options Written............................            61,292       2,198,065
Options Expired............................           (44,817)     (1,840,200)
Options Exercised..........................           (14,994)       (728,860)
Options Closed.............................            (4,614)       (303,035)
                                                    ---------      ----------
Options outstanding at May 31, 2012........            26,393      $1,061,285
                                                    =========      ==========


                                 5. BORROWINGS

The Fund has a committed facility agreement with the Bank of Nova Scotia
("Scotia") that has a maximum commitment amount of $145,000,000. 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, 2012 was $117,841,530, with a weighted average interest
rate of 0.91%. As of May 31, 2012, the Fund had outstanding borrowings of
$123,000,000 under this committed facility agreement. The high and low annual
interest rates for the six months ended May 31, 2012 were 0.95% and 0.89%,
respectively. The interest rate at May 31, 2012 was 0.89%.

The Fund's Board of Trustees, at a special meeting on March 27, 2012, approved
sub-advisor authority to enter into swap transactions under an ISDA
(International Swaps and Derivatives) Master Agreement with Credit Suisse
International. The swap transactions may be used to hedge against interest rate
risk on leverage, in addition the Fund already has standing permission to use
derivatives for investment purposes.

                               6. 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.

                                                                         Page 19



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NOTES TO FINANCIAL STATEMENTS - (CONTINUED)
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                     FIRST TRUST ENERGY INFRASTRUCTURE FUND
                            MAY 31, 2012 (UNAUDITED)


                             7. 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.

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.

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.

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.

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.

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.

                              8. SUBSEQUENT EVENTS

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

On June 20, 2012, the Fund declared a dividend of $0.1085 per share to Common
Shareholders of record on July 5, 2012, payable July 16, 2012.

On July 19, 2012, the Fund declared a dividend of $0.1085 per share to Common
Shareholders of record on August 3, 2012, payable August 15, 2012.



Page 20



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

                     FIRST TRUST ENERGY INFRASTRUCTURE FUND
                            MAY 31, 2012 (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



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

                     FIRST TRUST ENERGY INFRASTRUCTURE FUND
                            MAY 31, 2012 (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 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 Active Dividend Income Fund, First Trust Energy Infrastructure
Fund, Macquarie/First Trust Global Infrastructure/Utilities Dividend & Income
Fund and First Trust High Income Long/Short Fund was held on April 18, 2012 (the
"Annual Meeting"). At the Annual Meeting, Richard E. Erickson and Thomas R.
Kadlec were elected by the Common Shareholders of the First Trust Energy
Infrastructure Fund as Class II Trustees for a three-year term expiring at the
Fund's annual meeting of shareholders in 2015. The number of votes cast in favor
of Mr. Erickson was 16,564,990, the number of votes against was 219,218 and the
number of abstentions was 766,028. The number of votes cast in favor of Mr.
Kadlec was 16,592,516, the number of votes against was 191,692 and the number of
abstentions was 766,028. James A. Bowen, Niel B. Nielson and Robert F. Keith are
the other current and continuing Trustees.




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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
1 Wall Street
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 19, 2012
     ---------------------

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 19, 2012
     ---------------------

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

Date  July 19, 2012
     ---------------------

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