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Math Help - trouble starting integration by parts

  1. #1
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    trouble starting integration by parts

    xtan^2(x)dx on [0, pi/3]. i don't know how to get this one going. the book says to use substitution before integration by parts, but i don't see what to substitute. i tried 1+sec^2(x) for tan^2(x) but i didn't get anywhere.
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  2. #2
    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by vassago View Post
    xtan^2(x)dx on [0, pi/3]. i don't know how to get this one going. the book says to use substitution before integration by parts, but i don't see what to substitute. i tried 1+sec^2(x) for tan^2(x) but i didn't get anywhere.
    It should be \sec^2 x-1=\tan^2 x

    The integral would become \int_0^{\frac{\pi}{3}} \left(x\sec^2x-x\right)\,dx=\int_0^{\frac{\pi}{3}} x\sec^2x\,dx-\int_0^{\frac{\pi}{3}} x\,dx

    The first integral can then be done by parts, with u=x and \,dv=\sec^2x\,dx

    Can you take it from here?
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  3. #3
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    Quote Originally Posted by vassago View Post
    xtan^2(x)dx on [0, pi/3]. i don't know how to get this one going. the book says to use substitution before integration by parts, but i don't see what to substitute. i tried 1+sec^2(x) for tan^2(x) but i didn't get anywhere.
    Substitute \tan^2 x = \sec^2 x - 1. Then your integral becomes \int x \sec^2 x \, dx - \int x \, dx.

    Now use integration by parts on the first integral: u = x \Rightarrow du = dx and dv = \sec^2 x \, dx \Rightarrow v = \tan x.

    Blue sky all the way now.
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