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Math Help - Integration by subs. or parts?

  1. #1
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    Integration by subs. or parts?

    I have this question in a quiz for uni:



    They show the exact same question in my lecture notes equalling:

    1/2 e^x^2 + C , though im confused on how they get rid of the x in front: xe^x^2

    any help would be appreciated
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  2. #2
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by sterps
    I have this question in a quiz for uni:



    They show the exact same question in my lecture notes equalling:

    1/2 e^x^2 + C , though im confused on how they get rid of the x in front: xe^x^2

    any help would be appreciated
    \int dx \, xe^{x^2}

    Let y = x^2, then dy = 2x dx

    Then
    \int dx \, xe^{x^2} = \int \frac{dy}{2}e^y

    = \frac{1}{2}e^y + C = \frac{1}{2}e^{x^2} + C

    I don't know of a way to do this using integration by parts because \int dx \, e^{x^2} can't be done exactly.

    -Dan
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  3. #3
    Grand Panjandrum
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    You can do this without needing to integrate.

    Just differentiate (a) through (d), one of these derivatives will be the
    integrand, and that will be the correct solution.

    RonL
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  4. #4
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    Quote Originally Posted by CaptainBlack
    You can do this without needing to integrate.

    Just differentiate (a) through (d), one of these derivatives will be the
    integrand, and that will be the correct solution.

    RonL
    Yes, but that doesn't help integrating techniques much
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  5. #5
    Grand Panjandrum
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    Quote Originally Posted by Jameson
    Yes, but that doesn't help integrating techniques much
    It does if it teaches the student that they are allowed to spot the derivative
    under the integral sign, or to use the fundamental theorem where they might
    not have thought of it.

    RonL
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  6. #6
    Forum Admin topsquark's Avatar
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    And solving the problem backwards is always a good trick to have when doing multiple choice tests.


    -Dan
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