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Math Help - Insane Integrals

  1. #1
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    Insane Integrals

    Stuff like this makes me want to drop out.

    \int\sin^5 (4x) \cos^2 (4x)dx

    I think I would be fine if it weren't for those coefficients in there.

    Thanks
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  2. #2
    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by Chicken Enchilada View Post
    Stuff like this makes me want to drop out.

    \int\sin^5 (4x) \cos^2 (4x)dx

    I think I would be fine if it weren't for those coefficients in there.

    Thanks
    Since sine is raise to an odd power, break off a factor of sine:

    \int\sin^5\!\left(4x\right)\cos^2\!\left(4x\right)  \,dx=\int\left[\sin^2\!\left(4x\right)\right]^2\cos^2\!\left(4x\right)\sin\!\left(4x\right)\,dx

    Now apply the identity \sin^2u=1-\cos^2u to get:

    \int\left[1-\cos^2\!\left(4x\right)\right]^2\cos^2\!\left(4x\right)\sin\!\left(4x\right)\,dx.

    Now let u=\cos\!\left(4x\right)\implies -\tfrac{1}{4}\,du=\sin\!\left(4x\right)\,dx.

    Can you take it from here?
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