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Math Help - integration by parts with large trig functions

  1. #1
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    Question integration by parts with large trig functions

    my problem is to integrate (sin^5(x))(cos^18(x)). Lower limit is 0 and upper limit is pi/2. I tried graphing this and using calculators because my teacher didn't explain that far into using integration by parts.....
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    Quote Originally Posted by gogogo View Post
    my problem is to integrate (sin^5(x))(cos^18(x)). Lower limit is 0 and upper limit is pi/2. I tried graphing this and using calculators because my teacher didn't explain that far into using integration by parts.....
    straight substitution will work ...

    -\int_0^{\frac{\pi}{2}} \sin^4{x} \cdot \cos^{18}{x}(-\sin{x}) \, dx

    -\int_0^{\frac{\pi}{2}} (1-\cos^2{x})^2 \cdot \cos^{18}{x}(-\sin{x}) \, <br />
dx

    let u = \cos{x}

    du = -\sin{x} dx

    -\int_1^{0} (1-u^2)^2 \cdot u^{18} \, du

    \int_0^1 u^{18} - 2u^{20} + u^{22} \, du
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    wooooow! thanks!!!
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