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

                                    FORM N-Q

             QUARTERLY SCHEDULE OF PORTFOLIO HOLDINGS OF REGISTERED
                         MANAGEMENT INVESTMENT COMPANY

                  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: August 31, 2013
                                            -----------------

Form N-Q is to be used by management investment companies, other than small
business investment companies registered on Form N-5 (ss.ss. 239.24 and 274.5 of
this chapter), to file reports with the Commission, not later than 60 days after
the close of the first and third fiscal quarters, pursuant to rule 30b1-5 under
the Investment Company Act of 1940 (17 CFR 270.30b1-5). The Commission may use
the information provided on Form N-Q in its regulatory, disclosure review,
inspection, and policymaking roles.

A registrant is required to disclose the information specified by Form N-Q, and
the Commission will make this information public. A registrant is not required
to respond to the collection of information contained in Form N-Q 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 the
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. SCHEDULE OF INVESTMENTS. The Schedule(s) of Investments is attached
herewith.

FIRST TRUST ENERGY INFRASTRUCTURE FUND (FIF)
PORTFOLIO OF INVESTMENTS
AUGUST 31, 2013 (UNAUDITED)



    SHARES/
     UNITS                                    DESCRIPTION                                      VALUE
---------------  --------------------------------------------------------------------      --------------
COMMON STOCKS - 97.8%

                                                                                     
                 ELECTRIC UTILITIES - 28.8%
        285,800  Duke Energy Corp. (a).................................................    $   18,748,480
        224,400  Emera, Inc. (CAD) (a).................................................         6,355,124
        338,650  Exelon Corp...........................................................        10,325,438
        185,800  Fortis, Inc. (CAD) (a)................................................         5,353,679
        108,600  ITC Holdings Corp. (a)................................................         9,653,454
        273,300  NextEra Energy, Inc...................................................        21,962,388
        403,600  Northeast Utilities (a)...............................................        16,535,492
        147,400  NRG Yield, Inc., Class A (a)..........................................         4,106,564
        636,900  Southern (The) Co. (a)................................................        26,507,778
                                                                                           --------------
                                                                                              119,548,397
                                                                                           --------------

                 GAS UTILITIES - 7.7%
         50,800  Atmos Energy Corp. (a)................................................         2,049,780
        119,385  Laclede Group, Inc. (a)...............................................         5,316,214
        375,500  Questar Corp..........................................................         8,230,960
        415,479  UGI Corp. (a).........................................................        16,286,777
                                                                                           --------------
                                                                                               31,883,731
                                                                                           --------------

                 MULTI-UTILITIES - 15.8%
        103,200  ATCO, Ltd. (CAD) (a)..................................................         4,243,418
        149,000  Canadian Utilities, Ltd. (CAD) (a)....................................         4,833,694
        366,900  Dominion Resources, Inc...............................................        21,408,615
        237,000  National Grid PLC, ADR (a)............................................        13,641,720
        257,800  NiSource, Inc.........................................................         7,543,228
         34,000  Sempra Energy (a).....................................................         2,870,280
        265,100  Wisconsin Energy Corp. (a)............................................        10,879,704
                                                                                          _______________
                                                                                               65,420,659
                                                                                          _______________

                 OIL, GAS & CONSUMABLE FUELS - 45.5%
      1,454,550  Enbridge Energy Management, LLC (a) (b)...............................        43,621,954
        508,600  Enbridge Income Fund Holdings, Inc. (CAD) (a).........................        11,231,402
        278,569  Enbridge, Inc. (a)....................................................        11,407,401
         49,300  Keyera Corp. (CAD) (a)................................................         2,697,388
        538,403  Kinder Morgan Management, LLC (a) (b).................................        42,986,096
        616,500  Kinder Morgan, Inc. (a)...............................................        23,383,845
        222,100  Pembina Pipeline Corp. (CAD) (a)......................................         6,890,941
        401,300  Spectra Energy Corp. (a)..............................................        13,287,043
        342,300  TransCanada Corp. (a).................................................        14,931,126
        511,900  Williams (The) Cos., Inc..............................................        18,551,256
                                                                                           --------------
                                                                                              188,988,452
                                                                                           --------------
                 TOTAL COMMON STOCKS...................................................       405,841,239
                 (Cost $389,197,748)                                                       --------------

MASTER LIMITED PARTNERSHIPS - 32.6%

                 GAS UTILITIES - 3.0%
        172,764  AmeriGas Partners, L.P. (a)...........................................         7,342,470
        110,000  Suburban Propane Partners, L.P. (a)...................................         5,038,000
                                                                                           --------------
                                                                                               12,380,470
                                                                                          _______________


                See Notes to Quarterly Portfolio of Investments           Page 1



FIRST TRUST ENERGY INFRASTRUCTURE FUND (FIF)
PORTFOLIO OF INVESTMENTS (CONTINUED)
AUGUST 31, 2013 (UNAUDITED)



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

                                                                                     
                 INDEPENDENT POWER PRODUCERS & ENERGY TRADERS - 0.0%
          5,000  Brookfield Renewable Energy Partners, L.P. (CAD)......................    $      129,355
                                                                                           --------------

                 OIL, GAS & CONSUMABLE FUELS - 29.6%
         37,000  Access Midstream Partners, L.P. (a)...................................         1,687,570
         47,000  Alliance GP Holdings, L.P. (a)........................................         2,887,680
        139,715  Alliance Resource Partners, L.P. (a)..................................        10,517,745
        187,200  El Paso Pipeline Partners, L.P. (a)...................................         7,811,856
        100,000  Energy Transfer Equity, L.P. (a)......................................         6,433,000
        176,400  Energy Transfer Partners, L.P. (a)....................................         9,044,028
        130,600  Enterprise Products Partners, L.P. (a)................................         7,760,252
        102,700  EQT Midstream Partners, L.P. (a)......................................         4,918,303
        168,876  Holly Energy Partners, L.P. (a).......................................         6,000,164
        112,600  Magellan Midstream Partners, L.P. (a).................................         6,109,676
        105,239  Natural Resource Partners, L.P. (a)...................................         2,063,737
        103,472  NGL Energy Partners, L.P. (a).........................................         3,195,216
        154,200  ONEOK Partners, L.P. (a)..............................................         7,646,778
         16,100  Phillips 66 Partners, L.P. (c)........................................           495,397
        129,886  Plains All American Pipeline, L.P. (a)................................         6,567,036
        205,500  Spectra Energy Partners, L.P. (a).....................................         8,565,240
        177,095  TC Pipelines, L.P. (a)................................................         8,557,231
        344,628  Teekay LNG Partners, L.P. (a).........................................        14,501,946
        190,362  TransMontaigne Partners, L.P. (a).....................................         7,875,276
                                                                                           --------------
                                                                                              122,638,131
                                                                                           --------------
                 TOTAL MASTER LIMITED PARTNERSHIPS.....................................       135,147,956
                 (Cost $108,923,971)                                                       --------------

                 TOTAL INVESTMENTS - 130.4%............................................       540,989,195
                 (Cost $498,121,719) (d)                                                   --------------

   NUMBER OF
   CONTRACTS                                  DESCRIPTION                                      VALUE
---------------  ----------------------------------------------------------------------    --------------
CALL OPTIONS WRITTEN - (0.3%)
                 Dominion Resources, Inc. Calls
            300  @ $60.00 due October 2013.............................................           (18,300)
          1,900  @  62.50 due January 2014.............................................          (133,000)
                                                                                           --------------
                                                                                                 (151,300)
                                                                                           --------------
                 Duke Energy Corp. Call
          1,700  @  72.50 due October 2013.............................................           (11,900)
                                                                                           --------------
                 Enbridge, Inc. Calls
            701  @  45.00 due January 2014.............................................           (31,545)
          1,075  @  50.00 due January 2014.............................................           (13,975)
                                                                                           --------------
                                                                                                  (45,520)
                                                                                           --------------
                 Exelon Corp. Calls
          1,200  @  33.00 due September 2013...........................................            (3,600)
            600  @  40.00 due October 2013.............................................            (1,800)
            350  @  33.00 due January 2014.............................................           (17,500)
                                                                                           --------------
                                                                                                  (22,900)
                                                                                           --------------


Page 2          See Notes to Quarterly Portfolio of Investments



FIRST TRUST ENERGY INFRASTRUCTURE FUND (FIF)
PORTFOLIO OF INVESTMENTS (CONTINUED)
AUGUST 31, 2013 (UNAUDITED)



   NUMBER OF
   CONTRACTS                                  DESCRIPTION                                      VALUE
---------------  ----------------------------------------------------------------------    --------------
CALL OPTIONS WRITTEN (CONTINUED)
                                                                                     
                 Kinder Morgan, Inc. Calls
            500  @ $40.00 due September 2013...........................................    $       (5,000)
          1,400  @  42.50 due September 2013...........................................            (2,800)
            960  @  42.50 due December 2013............................................           (24,480)
            800  @  42.50 due January 2014.............................................           (31,600)
                                                                                           --------------
                                                                                                  (63,880)
                                                                                           --------------
                 National Grid PLC, ADR Call
          1,000  @  60.00 due December 2013............................................           (70,000)
                                                                                           --------------
                 Nextera Energy, Inc. Calls
          1,500  @  85.00 due September 2013...........................................           (22,500)
            300  @  87.50 due December 2013............................................           (19,500)
                                                                                           --------------
                                                                                                  (42,000)
                                                                                           --------------
                 NiSource, Inc. Calls
          1,000  @  32.00 due September 2013...........................................            (5,000)
          1,200  @  32.00 due January 2014.............................................           (33,000)
                                                                                           --------------
                                                                                                  (38,000)
                                                                                           --------------
                 Northeast Utilities Calls
            900  @  45.00 due October 2013.............................................            (9,000)
            461  @  45.00 due January 2014.............................................           (17,518)
                                                                                           --------------
                                                                                                  (26,518)
                                                                                           --------------
                 Questar Corp. Calls
          1,500  @  25.00 due October 2013.............................................            (7,500)
            500  @  26.00 due October 2013.............................................            (2,500)
            791  @  27.00 due October 2013.............................................            (3,955)
            964  @  26.00 due January 2014.............................................            (9,640)
                                                                                           --------------
                                                                                                  (23,595)
                                                                                           --------------
                 Southern (The) Co. Calls
          2,000  @  43.00 due November 2013............................................           (82,000)
              5  @  47.00 due November 2013............................................               (10)
                                                                                           --------------
                                                                                                  (82,010)
                                                                                           --------------
                 Spectra Energy Corp. Calls
            669  @  33.00 due September 2013...........................................           (40,140)
            800  @  34.00 due September 2013...........................................           (17,600)
          1,194  @  36.00 due September 2013...........................................            (5,970)
          1,350  @  37.00 due September 2013...........................................            (4,050)
                                                                                           --------------
                                                                                                  (67,760)
                                                                                           --------------
                 TransCanada Corp. Calls
            892  @  45.00 due November 2013............................................           (71,360)
            687  @  50.00 due November 2013............................................            (3,435)
                                                                                           --------------
                                                                                                  (74,795)
                                                                                           --------------
                 UGI Corp. Call
          1,800  @  40.00 due January 2014.............................................          (279,000)
                                                                                           --------------
                 Williams (The) Cos., Inc. Calls
            352  @  36.00 due October 2013.............................................           (39,424)


                See Notes to Quarterly Portfolio of Investments           Page 3



FIRST TRUST ENERGY INFRASTRUCTURE FUND (FIF)
PORTFOLIO OF INVESTMENTS (CONTINUED)
AUGUST 31, 2013 (UNAUDITED)



   NUMBER OF
   CONTRACTS                                  DESCRIPTION                                      VALUE
---------------  ----------------------------------------------------------------------    --------------
CALL OPTIONS WRITTEN (CONTINUED)
                                                                                     
                 Williams (The) Cos., Inc. Calls (Continued)
          1,000  @ $38.00 due October 2013.............................................    $      (42,000)
            482  @  35.00 due November 2013............................................          (107,486)
            500  @  36.00 due November 2013............................................           (85,500)
            700  @  37.00 due November 2013............................................           (84,000)
                                                                                           --------------
                                                                                                 (358,410)
                                                                                           --------------
                 Wisconsin Energy Corp. Calls
          1,000  @  45.00 due October 2013.............................................           (25,000)
            500  @  45.00 due April 2014...............................................           (22,500)
                                                                                           --------------
                                                                                                  (47,500)
                                                                                           --------------
                 TOTAL CALL OPTIONS WRITTEN............................................        (1,405,088)
                 (Premiums received $1,925,874)                                            --------------

                 OUTSTANDING LOAN - (35.2%)............................................      (145,900,000)

                 NET OTHER ASSETS AND LIABILITIES - 5.1%...............................        21,123,790
                                                                                           --------------
                 NET ASSETS - 100.0%...................................................    $  414,807,897
                                                                                           ==============


---------------------------------------
(a)   All or a portion of this security serves as collateral on the outstanding
      loan.
(b)   Non-income producing security which pays in-kind distributions in the form
      of additional shares.
(c)   Non-income producing security.
(d)   Aggregate cost for financial reporting purposes, which approximates the
      aggregate cost for federal income tax purposes. As of August 31, 2013, the
      aggregate gross unrealized appreciation for all securities in which there
      was an excess of value over tax cost was $55,470,215 and the aggregate
      gross unrealized depreciation for all securities in which there was an
      excess of tax cost over value was $12,602,739.

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.

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

OPEN SWAP CONTRACT OUTSTANDING:



                                                                                             UNREALIZED
LONG TOTAL RETURN                                            NOTIONAL       EXPIRATION     APPRECIATION/
EQUITY SWAP             COUNTERPARTY        PAY RATE          AMOUNT           DATE        (DEPRECIATION)
---------------------------------------------------------------------------------------------------------
                                                                            
                       Credit Suisse    1 month LIBOR +
Inter Pipeline Fund      International    40 basis          11,172,035       02/24/14      $      128,712



Page 4          See Notes to Quarterly Portfolio of Investments



FIRST TRUST ENERGY INFRASTRUCTURE FUND (FIF)
PORTFOLIO OF INVESTMENTS (CONTINUED)
AUGUST 31, 2013 (UNAUDITED)

VALUATION INPUTS
A summary of the inputs used to value the Fund's investments as of August 31,
2013 is as follows (see Note 2A - Portfolio Valuation in the Notes to Quarterly
Portfolio of Investments):


                                                 ASSETS TABLE
                                                                                    LEVEL 2         LEVEL 3
                                                     TOTAL          LEVEL 1       SIGNIFICANT     SIGNIFICANT
                                                    VALUE AT         QUOTED        OBSERVABLE     UNOBSERVABLE
INVESTMENTS                                        8/31/2013         PRICES          INPUTS          INPUTS
-----------------------------------------------  --------------  --------------  --------------  --------------
                                                                                     
Common Stocks*.................................  $  405,841,239  $  405,841,239  $           --  $           --
Master Limited Partnerships *..................     135,147,956     135,147,956              --              --
                                                 --------------  --------------  --------------  --------------
Total Investments..............................     540,989,195     540,989,195              --              --
                                                 --------------  --------------  --------------  --------------
OTHER FINANCIAL INSTRUMENTS
Total Return Equity Swap**.....................         128,712              --         128,712              --
                                                 --------------  --------------  --------------  --------------

TOTAL..........................................  $  541,117,907  $  540,989,195  $      128,712  $           --
                                                 ==============  ==============  ==============  ==============

                                                LIABILITIES TABLE
                                                                                    LEVEL 2         LEVEL 3
                                                     TOTAL          LEVEL 1       SIGNIFICANT     SIGNIFICANT
                                                    VALUE AT         QUOTED        OBSERVABLE     UNOBSERVABLE
INVESTMENTS                                        8/31/2013         PRICES          INPUTS          INPUTS
-----------------------------------------------  --------------  --------------  --------------  --------------
Call Options Written...........................  $   (1,405,088) $   (1,405,088) $           --  $           --
                                                 ==============  ==============  ==============  ==============


*  See Portfolio of Investments for industry breakout.
** See Open Swap Contract Outstanding for contract detail.

All transfers in and out of the Levels during the period are assumed to be
transferred on the last day of the period at their current value. There were no
transfers between Levels at August 31, 2013.


                See Notes to Quarterly Portfolio of Investments           Page 5



NOTES TO QUARTERLY PORTFOLIO OF INVESTMENTS

                  FIRST TRUST ENERGY INFRASTRUCTURE FUND (FIF)
                          AUGUST 31, 2013 (UNAUDITED)


                                1. ORGANIZATION

First Trust Energy Infrastructure Fund (the "Fund") is a non-diversified,
closed-end management investment company organized as a Massachusetts business
trust on February 22, 2011, and is registered with the Securities and Exchange
Commission ("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").

                     2. VALUATION AND INVESTMENT PRACTICES

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 Fund's securities, will be valued as follows:

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

      Exchange-traded options 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.

      Swaps are valued utilizing quotations provided by a third party pricing
      service.

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

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

      1)    the type of security;

      2)    the size of the holding;

      3)    the initial cost of the security;

      4)    transactions in comparable securities;

      5)    price quotes from dealers and/or pricing services;

      6)    relationships among various securities;

      7)    information obtained by contacting the issuer, analysts, or the
            appropriate stock exchange;

      8)    an analysis of the issuer's financial statements; and

      9)    the existence of merger proposals or tender offers that might affect
            the value of the security.

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

      1)    the value of similar foreign securities traded on other foreign
            markets;

      2)    ADR trading of similar securities;

      3)    closed-end fund trading of similar securities;

      4)    foreign currency exchange activity;

Page 6



NOTES TO FINANCIAL STATEMENTS - (CONTINUED)

                  FIRST TRUST ENERGY INFRASTRUCTURE FUND (FIF)
                          AUGUST 31, 2013 (UNAUDITED)


      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 August 31, 2013, is
included with the Fund's Portfolio of Investments.

B. OPTION CONTRACTS:

The Fund is subject to equity price risk in the normal course of pursuing its
investment objective and may write (sell) options to hedge against changes in
the value of equities. Also, the Fund seeks to generate additional income, in
the form of premiums received, from writing (selling) the options. The Fund may
write (sell) covered call or put options ("options") on all or a portion of the
common stock 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. 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.

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

                                                                          Page 7



NOTES TO FINANCIAL STATEMENTS - (CONTINUED)

                  FIRST TRUST ENERGY INFRASTRUCTURE FUND (FIF)
                          AUGUST 31, 2013 (UNAUDITED)

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

C. SWAP AGREEMENTS:

The Fund may enter into swap agreements as a substitute for investing in certain
securities. A swap is a financial instrument that typically involves the
exchange of cash flows between two parties ("Counterparties") on specified dates
(settlement dates) where the cash flows are based on agreed-upon prices, rates,
etc. In a typical swap agreement, one party agrees to pay another party the
return on a security or basket of securities in return for payment of a
specified interest rate. By entering into swaps, the Fund can gain exposure to a
security without actually purchasing the underlying asset. Swap agreements
involve the risk associated with the investment in the security as well as the
risk that the performance of the security, including any dividends, will not
exceed the interest that the Fund will be committed to pay under the swap. Swap
agreements are individually negotiated and involve the risk of the potential
inability of the Counterparties to meet the terms of the agreement. In
connection with these agreements, cash and securities may be identified as
collateral in accordance with the terms of the respective swap agreements to
provide assets of value and recourse in the event of default under the swap
agreement or bankruptcy/insolvency of a party to the swap agreement. In the
event of a default by the Counterparty, the Fund will seek withdrawal of this
collateral and may incur certain costs exercising its right with respect to the
collateral. If a Counterparty becomes bankrupt or otherwise fails to perform its
obligations due to financial difficulties, the Fund may experience significant
delays in obtaining any recovery in a bankruptcy or other reorganization
proceeding. The Fund may obtain only limited recovery or may obtain no recovery
in such circumstances.

Swap agreements may increase or decrease the overall volatility of the
investments of the Fund. The performance of swap agreements may be affected by a
change in the specific interest rate, security, currency, or other factors that
determine the amounts of payments due to and from the Fund. A total return
equity swap agreement would expose the Fund to the same equity price risk as it
would have if the underlying equity securities were purchased. The Fund's
maximum equity price risk to meet its future payments under swap agreements
outstanding as of August 31, 2013, is equal to the total notional amount as
shown on the Portfolio of Investments. The notional amount represents the U.S.
dollar value of the contract as of the day of the opening transaction or
contract reset. The Fund entered into a Total Return Equity Swap agreement on
January 22, 2013. The average volume of Total Return Equity Swaps was $8,423,836
for the period January 22, 2013, to August 31, 2013.

D. SECURITIES TRANSACTIONS:

Securities transactions are recorded as of the trade date. Realized gains and
losses from securities transactions are recorded on the identified cost basis.

E. 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 investments and items of income and expense are
translated on the respective dates of such transactions.

                               3. OPTION ACTIVITY

Written option activity for the Fund for the fiscal year to date period
(December 1, 2012 through August 31, 2013) was as follows:

                                                   Number
                                                     of
Written Options                                   Contracts     Premiums
---------------------------------------------    ----------    -----------
Options outstanding at November 30, 2012...           35,234   $ 1,576,846
Options Written............................           81,816     3,962,046
Options Expired............................         (48,194)    (2,138,461)
Options Exercised..........................         (31,473)    (1,414,964)
Options Closed.............................          (1,850)       (59,593)
                                                 ----------    -----------
Options outstanding at August 31, 2013.....          35,533    $ 1,925,874
                                                 ==========    ===========

Page 8




ITEM 2. 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 last fiscal quarter
       that have materially affected, or are reasonably likely to materially
       affect, the registrant's internal control over financial reporting.

ITEM 3. EXHIBITS.

Certifications pursuant to Rule 30a-2(a) under the 1940 Act and Section 302 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: October 15, 2013
     ---------------------

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: October 15, 2013
     ---------------------

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

Date: October 15, 2013
     ---------------------

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