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Math Help - Definite integral with u substitution

  1. #1
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    Definite integral with u substitution

    Integral 1 to 2 of (x^3)lnxdx

    What do we use as u? I tried lnx and x^3 and it didnt seem to work.
    Last edited by Steelers72; March 4th 2013 at 03:01 PM.
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  2. #2
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    Re: Definite integral with u substitution

    The method is to integrate by parts.
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    Re: Definite integral with u substitution

    You should remember the acronym LIATE: you want to choose for u whatever comes first in the list Logarithmic, Inverse trigonometric, Algebraic, Trigonometric, Exponential. In this case you would choose u=\ln{x}.

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    Re: Definite integral with u substitution

    Quote Originally Posted by BobP View Post
    The method is to integrate by parts.
    Yeah my friend and I realized this when we looked on the page before it said with product do integration by parts haha...Thanks!
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    Re: Definite integral with u substitution

    Here's another method:

    \int_1^2 x^a dx=\frac{2^{1+a}-1}{1+a}

    Differentiate with respect to a:

    \int_1^2 x^a \log(x) dx=\frac{2^{1+a}\log(2)(1+a)-2^{1+a}+1}{(1+a)^2}

    Putting a=3, gives the answer

    \int_1^2 x^3 \log(x) dx= \frac{4\times 2^4 \log(2)-2^4+1}{4^2}=4\log(2)-\frac{15}{16}
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