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Math Help - Integrating by parts

  1. #1
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    Integrating by parts

    I have a problem that I've been working on:

    Consider the definite integral

    I have:

    arcsin(4x)(x^2/2) minus the integral [IMG]file:///C:/Users/Tawny/AppData/Local/Temp/moz-screenshot.jpg[/IMG][IMG]file:///C:/Users/Tawny/AppData/Local/Temp/moz-screenshot-1.jpg[/IMG][IMG]file:///C:/Users/Tawny/AppData/Local/Temp/moz-screenshot-2.jpg[/IMG]

    I'm really not sure where to go from here.
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  2. #2
    Super Member redsoxfan325's Avatar
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    Quote Originally Posted by C.C. View Post
    I have a problem that I've been working on:

    Consider the definite integral

    I have:

    arcsin(4x)(x^2/2) minus the integral [IMG]file:///C:/Users/Tawny/AppData/Local/Temp/moz-screenshot.jpg[/IMG][IMG]file:///C:/Users/Tawny/AppData/Local/Temp/moz-screenshot-1.jpg[/IMG][IMG]file:///C:/Users/Tawny/AppData/Local/Temp/moz-screenshot-2.jpg[/IMG]

    I'm really not sure where to go from here.
    If x=\frac{1}{4}\sin(u), then dx=\frac{1}{4}\cos(u)

    Since u=\arcsin(4x), the new bounds are 0...\frac{\pi}{2}

    So we have \frac{1}{16}\int_0^{\pi/2} u\sin u\cos u\,du=\frac{1}{32}\int_0^{\pi/2}u\sin2u\,du

    From here it's an easy integration by parts. I calculate the final answer to be \frac{\pi}{128}.
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