The Collaborative International Dictionary of English v.0.48:

Undertake \Un`der*take"\, v. t. [imp. Undertook; p. p.
   Undertaken; p. pr. & vb. n. Undertaking.] [Under + take.]
   1. To take upon one's self; to engage in; to enter upon; to
      take in hand; to begin to perform; to set about; to
      attempt.
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            To second, or oppose, or undertake
            The perilous attempt.                 --Milton.
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   2. Specifically, to take upon one's self solemnly or
      expressly; to lay one's self under obligation, or to enter
      into stipulations, to perform or to execute; to covenant;
      to contract.
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            I 'll undertake to land them on our coast. --Shak.
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   3. Hence, to guarantee; to promise; to affirm.
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            And he was not right fat, I undertake. --Dryden.
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            And those two counties I will undertake
            Your grace shall well and quietly enjoiy. --Shak.
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            I dare undertake they will not lose their labor.
                                                  --Woodward.
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   4. To assume, as a character. [Obs.] --Shak.
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   5. To engage with; to attack. [Obs.]
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            It is not fit your lordship should undertake every
            companion that you give offense to.   --Shak.
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   6. To have knowledge of; to hear. [Obs.] --Spenser.
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   7. To take or have the charge of. [Obs.] "Who undertakes you
      to your end." --Shak.
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            Keep well those that ye undertake.    --Chaucer.
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The Collaborative International Dictionary of English v.0.48:

Undertake \Un`der*take"\, v. i.
   1. To take upon one's self, or assume, any business, duty, or
      province.
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            O Lord, I am oppressed; undertake for me. --Isa.
                                                  xxxviii. 14.
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   2. To venture; to hazard. [Obs.]
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            It is the cowish terror of his spirit
            That dare not undertake.              --Shak.
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   3. To give a promise or guarantee; to be surety.
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            But on mine honor dare I undertake
            For good lord Titus' innocence in all. --Shak.
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WordNet (r) 3.0 (2006):

undertake
    v 1: enter upon an activity or enterprise [syn: undertake,
         set about, attempt]
    2: accept as a challenge; "I'll tackle this difficult task"
       [syn: undertake, tackle, take on]
    3: promise to do or accomplish; "guarantee to free the
       prisoners" [syn: undertake, guarantee]
    4: enter into a contractual arrangement [syn: contract,
       undertake]
    5: accept as a charge [syn: undertake, take in charge]