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Math Help - integration by parts

  1. #1
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    integration by parts

    use integration by parts to find:

    \int^{\frac{\pi}{2}}_{0} xsin^2xcosx dx
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  2. #2
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    Quote Originally Posted by qzno View Post
    use integration by parts to find:

    \int^{\frac{\pi}{2}}_{0} xsin^2xcosx dx
    u = x

    du = dx

    dv = \sin^2{x}\cos{x} \, dx

    v = \frac{\sin^3{x}}{3}

    \int x\sin^2{x}\cos{x} \, dx = \frac{x\sin^3{x}}{3} - \int \frac{\sin^3{x}}{3} \, dx

    for the last integral ...

    \sin^3{x} = (1 - \cos^2{x})\sin{x} = \sin{x} + \cos^2{x}(-\sin{x})
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    dv = \sin^2{x}\cos{x} \, dx

    v = \frac{\sin^3{x}}{3}

    is this a rule?
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  4. #4
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    Quote Originally Posted by qzno View Post
    dv = \sin^2{x}\cos{x} \, dx

    v = \frac{\sin^3{x}}{3}

    is this a rule?
    it's an antiderivative using informal substitution ... note that \cos{x} is the derivative of \sin{x} ...

    \sin^2{x}\cos{x} \, dx is the same as t^2 dt
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