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Math Help - need some explanation on integration

  1. #1
    Member Jones's Avatar
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    need some explanation on integration

    Hello,

    I'm trying to integrate the following:

    \int sin^4 x\cdot cos^5 x

    rewrite:
    sin^4x(cosx\cdot cos^4 x)

    cos^2 x = (1-sin^2x)

    cos^4 x = (1-sin^2x)^2 = 1-2sin^2x+sin^4x

    multiply together:

    sin^4x-2sin^6x+sin^8x \cdot cosx

    let u = sin x, du = cos x

    we have u^4-2u^6+u^8 du

    My question is, the derivative of u is cos x

    But we have 3 u terms. Aren't you getting a ŽduŽ for every u term?
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  2. #2
    Ted
    Ted is offline
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    Quote Originally Posted by Jones View Post
    Hello,

    I'm trying to integrate the following:

    \int sin^4 x\cdot cos^5 x

    rewrite:
    sin^4x(cosx\cdot cos^4 x)

    cos^2 x = (1-sin^2x)

    cos^4 x = (1-sin^2x)^2 = 1-2sin^2x+sin^4x

    multiply together:

    sin^4x-2sin^6x+sin^8x \cdot cosx

    let u = sin x, du = cos x

    we have u^4-2u^6+u^8 du

    My question is, the derivative of u is cos x

    But we have 3 u terms. Aren't you getting a ŽduŽ for every u term?
    Nope. only one du, here:

    Quote Originally Posted by Jones View Post
    sin^4x-2sin^6x+sin^8x \cdot cosx
    3
    You should have brackets here.

    It should be {\color{red} ( } sin^4(x)-2sin^6(x)+sin^8(x) {\color{red} ) } cos(x)

    take u=sin(x) to get:

    \int (u^4-2u^6+u^8)du which is easy.
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