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Math Help - Integration of trigonometric Identities.

  1. #1
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    Integration of trigonometric Identities.

    Problem:
    \int sinx(1-sin2x)^{3}dx


    I have been trying this calculus problem for so long and don't seem to solve it, so please help me solve it?



    Thanks for any help
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  2. #2
    MHF Contributor red_dog's Avatar
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    (1-\sin 2x)^3=(1-2\sin x\cos x)^3=1-6\sin x\cos x+12\sin^2x\cos^2x-8\sin^3x\cos^3x

    Then

    \int\sin x(1-\sin 2x)^3dx=\int(\sin x-6\sin^2x\cos x+12\sin^3x\cos^2x-8\sin^4x\cos^3x)dx=

    =\int\sin xdx-6\int\sin^2x\cos xdx+

    +12\int(1-\cos^2x)\cos^2x\sin xdx-8\int\sin^4x(1-\sin^2x)\cos xdx

    The first integral is easy.

    In the second let u-\sin x

    In the third let u=\cos x

    In the fourth let u=\sin x
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