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]

Moby Thesaurus II by Grady Ward, 1.0:

115 Moby Thesaurus words for "undertake":
   accept, accept obligation, agree, agree to, answer for, approach,
   assay, assume, attack, attempt, bargain, bargain for,
   be answerable for, be responsible for, be security for, begin,
   bind, buckle to, carry on, certify, chance, commence, commit,
   compact, conduct, contract, covenant, dive into, do, do a deal,
   embark in, embark on, embark upon, employ, endeavor, engage,
   engage in, enter on, enter upon, essay, exercise, fall into,
   fall to, follow, get busy, get cracking, get going, get under way,
   get with it, go about, go at, go bail for, go in for, go into,
   go to it, go upon, guarantee, have an understanding, have at,
   hop to it, jump to it, launch forth, launch into, lay about,
   lift a finger, light into, make a deal, make an attempt,
   make an effort, move into, obligate, offer, pass, pitch in,
   pitch into, pledge, plunge into, practice, proceed to, promise,
   prosecute, pursue, sail into, seek, set about, set at, set forward,
   set going, set to, set to work, shake hands on, specialize in,
   stand back of, start, start in, stipulate, strive, struggle, swear,
   tackle, take on, take the vows, take to, take up, try, turn to,
   use, venture, venture on, venture upon, vow, wade into, wage,
   warrant, work at