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
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               (Exact name of registrant as specified in charter)

                       120 East Liberty Drive, Suite 400
                               Wheaton, IL 60187
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              (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
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                    (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: February 28, 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
PORTFOLIO OF INVESTMENTS
FEBRUARY 28, 2013 (UNAUDITED)



    SHARES/
     UNITS                                      DESCRIPTION                                    VALUE
---------------  --------------------------------------------------------------------      --------------
COMMON STOCKS - 98.0%

                 ELECTRIC UTILITIES - 28.3%
                                                                                     
        263,266  Duke Energy Corp. (a)...............................................      $   18,231,170
        254,600  Emera, Inc. (CAD) (a)...............................................           8,749,599
        366,450  Exelon Corp. (a)....................................................          11,356,286
        208,800  Fortis, Inc. (CAD) (a)..............................................           6,815,232
        105,000  ITC Holdings Corp. (a)..............................................           8,874,600
        294,400  NextEra Energy, Inc. (a)............................................          21,158,528
        351,600  Northeast Utilities (a).............................................          14,594,916
        595,500  Southern (The) Co. (a)..............................................          26,803,455
                                                                                           --------------
                                                                                              116,583,786
                                                                                           --------------

                 GAS UTILITIES - 6.1%
        403,900  Questar Corp. (a)...................................................           9,495,689
        436,479  UGI Corp. (a).......................................................          15,634,678
                                                                                           --------------
                                                                                               25,130,367
                                                                                           --------------

                 MULTI-UTILITIES - 18.3%
         35,000  ATCO, Ltd., Class I (CAD) (a).......................................           3,024,000
         45,500  Canadian Utilities, Ltd. (CAD) (a)..................................           3,430,865
        520,600  Centerpoint Energy, Inc. (a)........................................          11,156,458
        363,900  Dominion Resources, Inc. ...........................................          20,378,400
        244,000  National Grid PLC, ADR (a)..........................................          13,444,400
        495,400  NiSource, Inc. (a)..................................................          13,722,580
         34,000  Sempra Energy ......................................................           2,643,840
        185,000  Wisconsin Energy Corp. (a)..........................................           7,640,500
                                                                                           --------------
                                                                                               75,441,043
                                                                                           --------------

                 OIL, GAS & CONSUMABLE FUELS - 45.3%
      1,371,427  Enbridge Energy Management, LLC (a) (b).............................          37,481,112
        585,600  Enbridge Income Fund Holdings, Inc. (CAD) (a).......................          14,298,577
        304,869  Enbridge, Inc. (a)..................................................          13,584,963
        157,300  Keyera Corp. (CAD) (a)..............................................           8,305,440
        507,565  Kinder Morgan Management, LLC (a) (b)...............................          42,041,609
        286,500  Kinder Morgan, Inc. ................................................          10,620,555
        307,100  Pembina Pipeline Corp. (CAD) (a)....................................           8,618,157
        560,950  Spectra Energy Corp. (a)............................................          16,289,988
        335,400  TransCanada Corp. (a)...............................................          15,599,454
        560,700  Williams (The) Cos., Inc. (a).......................................          19,461,897
                                                                                           --------------
                                                                                              186,301,752
                                                                                           --------------
                 TOTAL COMMON STOCKS ................................................         403,456,948
                 (Cost $361,478,823)                                                       --------------

MASTER LIMITED PARTNERSHIPS - 33.1%

                 GAS UTILITIES - 3.8%
        232,764  AmeriGas Partners, L.P. (a).........................................          10,169,459
        125,000  Suburban Propane Partners, L.P. (a).................................           5,262,500
                                                                                           --------------
                                                                                               15,431,959
                                                                                           --------------

                 OIL, GAS & CONSUMABLE FUELS - 29.3%
         70,000  Access Midstream Partners, L.P. (a).................................           2,606,800


                See Notes to Quarterly Portfolio of Investments           Page 1



FIRST TRUST ENERGY INFRASTRUCTURE FUND
PORTFOLIO OF INVESTMENTS  - (CONTINUED)
FEBRUARY 28, 2013 (UNAUDITED)



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

                 OIL, GAS & CONSUMABLE FUELS - (CONTINUED)
                                                                                     
         47,000  Alliance GP Holdings, L.P. (a)......................................      $    2,441,180
         81,465  Alliance Resource Partners, L.P. (a)................................           5,058,162
        142,000  El Paso Pipeline Partners, L.P. (a).................................           5,934,180
        149,000  Energy Transfer Equity, L.P. (a)....................................           7,925,310
         91,600  Enterprise Products Partners, L.P. (a)..............................           5,190,972
         43,700  EQT Midstream Partners, L.P. (a)....................................           1,657,978
        226,446  Holly Energy Partners, L.P. (a).....................................           9,343,162
        103,600  Magellan Midstream Partners, L.P. (a)...............................           5,196,576
         93,400  MPLX, L.P. (a)......................................................           3,053,246
        143,239  Natural Resource Partners, L.P. (a).................................           3,132,637
        242,000  NuStar Energy, L.P. (a).............................................          12,351,680
         29,600  NuStar GP Holdings, LLC (a).........................................             895,400
        126,000  ONEOK Partners, L.P. (a)............................................           6,906,060
        148,886  Plains All American Pipeline, L.P. (a)..............................           8,151,508
        124,900  Spectra Energy Partners, L.P. (a)...................................           4,611,308
         21,709  Sunoco Logistics Partners, L.P. (a).................................           1,357,464
        253,095  TC Pipelines, L.P. (a)..............................................          11,606,937
        306,328  Teekay LNG Partners, L.P. (a).......................................          11,992,741
        146,570  TransMontaigne Partners, L.P. (a)...................................           6,721,700
         91,300  Williams Partners, L.P. (a).........................................           4,537,610
                                                                                           --------------
                                                                                              120,672,611
                                                                                           --------------
                 TOTAL MASTER LIMITED PARTNERSHIPS ..................................         136,104,570
                 (Cost $109,936,046)                                                       --------------

                 TOTAL INVESTMENTS - 131.1% .........................................         539,561,518
                 (Cost $471,414,869) (c)                                                   --------------

   NUMBER OF
   CONTRACTS                                 DESCRIPTION                                       VALUE
---------------  --------------------------------------------------------------------      --------------
CALL OPTIONS WRITTEN - (0.8%)
                 Centerpoint Energy, Inc. Call
            400  @  $22.50 due May 2013..............................................              (9,600)
                                                                                           --------------
                 Dominion Resources, Inc. Calls
          3,369  @   55.00 due April 2013............................................            (555,885)
            270  @   50.00 due July 2013.............................................            (171,450)
                                                                                           --------------
                                                                                                 (727,335)
                                                                                           --------------
                 Duke Energy Corp. Call
          2,391  @   70.00 due April 2013............................................            (191,280)
                                                                                           --------------
                 Enbridge, Inc. Calls
            859  @   45.00 due April 2013............................................             (57,982)
            700  @   45.00 due July 2013.............................................             (84,000)
                                                                                           --------------
                                                                                                 (141,982)
                                                                                           --------------
                 Exelon Corp. Calls
            500  @   31.00 due April 2013............................................             (40,000)


Page 2          See Notes to Quarterly Portfolio of Investments



FIRST TRUST ENERGY INFRASTRUCTURE FUND
PORTFOLIO OF INVESTMENTS  - (CONTINUED)
FEBRUARY 28, 2013 (UNAUDITED)



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

                 Exelon Corp. Calls (Continued)
                                                                                     
            820  @  $32.00 due April 2013............................................      $      (28,700)
                                                                                           --------------
                                                                                                  (68,700)
                                                                                           --------------
                 Kinder Morgan, Inc. Calls
          1,264  @   40.00 due March 2013............................................              (6,320)
          1,600  @   40.00 due June 2013.............................................             (76,800)
                                                                                           --------------
                                                                                                  (83,120)
                                                                                           --------------
                 National Grid PLC, ADR Call
            200  @   60.00 due March 2013............................................              (1,000)
                                                                                           --------------
                 Nextera Energy, Inc. Call
          2,125  @   75.00 due June 2013.............................................            (116,875)
                                                                                           --------------
                 Northeast Utilities Call
          1,433  @   40.00 due April 2013............................................            (186,290)
                                                                                           --------------
                 Questar Corp. Calls
            574  @   23.00 due March 2013............................................             (41,615)
            457  @   21.00 due April 2013............................................            (122,248)
            750  @   22.00 due April 2013............................................            (133,125)
          1,000  @   21.00 due July 2013.............................................            (285,000)
                                                                                           --------------
                                                                                                 (581,988)
                                                                                           --------------
                 Southern (The) Co. Calls
          1,000  @   46.00 due May 2013..............................................             (38,000)
          2,422  @   47.00 due August 2013...........................................             (99,302)
                                                                                           --------------
                                                                                                 (137,302)
                                                                                           --------------
                 Spectra Energy Corp. Calls
          1,000  @   31.00 due March 2013............................................              (7,500)
          1,000  @   29.00 due March 2013............................................             (50,000)
          2,580  @   29.00 due June 2013.............................................            (309,600)
                                                                                           --------------
                                                                                                 (367,100)
                                                                                           --------------
                 TransCanada Corp. Calls
          1,687  @   50.00 due May 2013..............................................             (42,175)
            333  @   55.00 due August 2013...........................................              (4,995)
                                                                                           --------------
                                                                                                  (47,170)
                                                                                           --------------
                 UGI Corp. Call
          1,000  @   35.00 due April 2013............................................            (107,500)
                                                                                           --------------
                 Williams (The) Cos., Inc. Calls
          1,500  @   36.00 due March 2013............................................             (16,500)
          1,507  @   34.00 due May 2013..............................................            (236,599)
          1,500  @   37.00 due May 2013..............................................             (70,500)
                                                                                           --------------
                                                                                                 (323,599)
                                                                                           --------------


                See Notes to Quarterly Portfolio of Investments           Page 3



FIRST TRUST ENERGY INFRASTRUCTURE FUND
PORTFOLIO OF INVESTMENTS  - (CONTINUED)
FEBRUARY 28, 2013 (UNAUDITED)



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

                 Wisconsin Energy Corp. Call
                                                                                     
          1,000  @  $40.00 due July 2013.............................................      $     (176,000)
                                                                                           --------------
                 TOTAL CALL OPTIONS WRITTEN .........................................          (3,266,841)
                 (Premiums received $1,533,446)                                            --------------


                 OUTSTANDING LOAN - (34.5%) .........................................        (141,900,000)

                 NET OTHER ASSETS AND LIABILITIES - 4.2% ............................          17,067,719
                                                                                           --------------
                 NET ASSETS - 100.0% ................................................      $  411,462,396
                                                                                           ==============


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

(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.

(c)   Aggregate cost for financial reporting purposes, which approximates the
      aggregate cost for federal income tax purposes. As of February 28, 2013,
      the aggregate gross unrealized appreciation for all securities in which
      there was an excess of value over tax cost was $72,954,663 and the
      aggregate gross unrealized depreciation for all securities in which there
      was an excess of tax cost over value was $4,808,014.

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.



                                                                                             UNREALIZED
LONG TOTAL RETURN                                            NOTIONAL       EXPIRATION     APPRECIATION/
EQUITY SWAP             COUNTERPARTY        PAY RATE          AMOUNT           DATE        (DEPRECIATION)
---------------------------------------------------------------------------------------------------------
                                                                               
                       Credit Suisse   1 month LIBOR +40
Inter Pipeline Fund    International      basis points      $ 5,290,955     02/24/14         $ (85,492)



Page 4          See Notes to Quarterly Portfolio of Investments



FIRST TRUST ENERGY INFRASTRUCTURE FUND
PORTFOLIO OF INVESTMENTS  - (CONTINUED)
FEBRUARY 28, 2013 (UNAUDITED)

VALUATION INPUTS

A summary of the inputs used to value the Fund's investments as of February 28,
2013 is as follows (see Note 1A - 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                                        2/28/2013         PRICES          INPUTS          INPUTS
-----------------------------------------------  --------------  --------------  --------------  --------------
                                                                                     
Common Stocks*.................................  $  403,456,948  $  403,456,948  $           --  $           --
Master Limited Partnerships*...................     136,104,570     136,104,570              --              --
                                                 --------------  --------------  --------------  --------------
TOTAL INVESTMENTS..............................  $  539,561,518  $  539,561,518  $           --  $           --
                                                 ==============  ==============  ==============  ==============

* See Portfolio of Investments for industry breakout.

                                               LIABILITIES TABLE
                                                                                    LEVEL 2         LEVEL 3
                                                     TOTAL          LEVEL 1       SIGNIFICANT     SIGNIFICANT
                                                   VALUE AT         QUOTED        OBSERVABLE     UNOBSERVABLE
                                                   2/28/2013        PRICES          INPUTS          INPUTS
-----------------------------------------------  --------------  --------------  --------------  --------------
Call Options Written...........................  $   (3,266,841) $   (3,266,841) $           --  $           --
Total Return Equity Swap.......................         (85,492)             --         (85,492)             --
                                                 --------------  --------------  --------------  --------------
TOTAL..........................................  $   (3,352,333) $   (3,266,841) $      (85,492) $           --
                                                 ==============  ==============  ==============  ==============



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 February 28, 2013.

                See Notes to Quarterly Portfolio of Investments           Page 5



NOTES TO QUARTERLY PORTFOLIO OF INVESTMENTS

                     FIRST TRUST ENERGY INFRASTRUCTURE FUND
                         FEBRUARY 28, 2013 (UNAUDITED)



                     1. VALUATION AND INVESTMENT PRACTICES

A. PORTFOLIO VALUATION:

The net asset value ("NAV") of the Common Shares of First Trust Energy
Infrastructure Fund (the "Fund") is determined daily as of the close of regular
trading on the New York Stock Exchange ("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 Investment Company Act of 1940, as amended (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.

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;

      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.

Page 6



NOTES TO FINANCIAL STATEMENTS - (CONTINUED)

                     FIRST TRUST ENERGY INFRASTRUCTURE FUND
                         FEBRUARY 28, 2013 (UNAUDITED)

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 February 28, 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 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. 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
                         FEBRUARY 28, 2013 (UNAUDITED)

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. 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 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 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 both 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. Swaps are individually negotiated. 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
February 28, 2013 is equal to the total notional amount as shown on the
Portfolio of Investments. The Fund entered into Total Return Equity Swap
agreements on January 22, 2013. The average volume of Total Return Equity Swaps
was $5,290,955 for the period January 22, 2013 to February 28, 2013. The
notional amount represents the U.S. value of the contracts as of the day of the
opening transaction.

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.

                               2. OPTION ACTIVITY

Written option activity for the Fund for the period ended February 28, 2013 was
as follows:

                                                   Number
                                                     of
Written Options                                   Contracts     Premiums
---------------------------------------------    ----------    -----------
Options outstanding at November 30, 2012.....      35,234      $ 1,576,846
Options Written..............................      26,013        1,182,975
Options Expired..............................     (20,071)        (954,738)
Options Exercised............................      (5,935)        (271,637)
Options Closed...............................          --               --
                                                 ----------    -----------
Options outstanding at February 28, 2013.....      35,241      $ 1,533,446
                                                 ==========    ===========

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  April 12, 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  April 12, 2013
     -------------------

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

Date  April 12, 2013
     -------------------

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