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Math Help - Integral xsin^2(x)cos(x)ds

  1. #1
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    Integral xsin^2(x)cos(x)ds

    I tried u=sin(x)
    du=cos(x)dx
    but that doesn't work
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  2. #2
    MHF Contributor Mathstud28's Avatar
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    Quote Originally Posted by khuezy View Post
    I tried u=sin(x)
    du=cos(x)dx
    but that doesn't work
    This isnt fun

    Seeing that \sin^2(x)=1-\cos^2(x)

    we have \int{x\cos(x)-x\cos^3(x)}dx

    Use parts

    and on the second one see that \cos^3(x)=\cos(x)(1-\sin^2(x))
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  3. #3
    Super Member PaulRS's Avatar
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    Let: \int_0^x\sin^2(t)\cos(t)dt=\int_0^{\sin(x)}u^2du=\  frac{\sin^3(x)}{3}=P(x)

    \int{x\sin^2(x)\cos(x)dx}=\int{x\cdot{P'(x)}dx} now apply parts

    \int{x\cdot{P'(x)}dx}=x\cdot{P(x)}-\int{P(x)dx}

    And remember that: \sin(3x)=-4\sin^3(x)+3\sin(x) for your last integral
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