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Math Help - Integral powers of sine and cosine

  1. #1
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    Exclamation Integral powers of sine and cosine

    Question :

    \int sin^4.cos^2x.dx

    Attempt :
    I reached a step where >>

    \frac{1}{8} \int (1-2cos2x+cos^2x).(1+cos2x)

    How to expand this ?
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  2. #2
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    Quote Originally Posted by mj.alawami View Post
    Question :

    \int sin^4.cos^2x.dx

    Attempt :
    I reached a step where >>

    \frac{1}{8} \int (1-2cos2x+cos^2x).(1+cos2x)

    How to expand this ?
    By the same method of expanding (a+b+c)(d+e).
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  3. #3
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    Quote Originally Posted by mj.alawami View Post
    Question :

    \int sin^4.cos^2x.dx

    Attempt :
    I reached a step where >>

    \frac{1}{8} \int (1-2cos2x+cos^2x).(1+cos2x)

    How to expand this ?
    \int{sin^4xcos^2x}dx=\int{sin^2xsin^2xcos^2x}dx

    =\int{\left(1-cos^2x\right)\left(1-cos^2x\right)cos^2x}dx

    =\int{\left(1-2cos^2x+cos^4x\right)cos^2x}dx

    may be more straightforward.

    Also you have erred on the way...

    sin^2xsin^2x=\frac{1}{4}(1-cos2x)(1-cos2x)=\frac{1}{4}\left(1-2cos2x+cos^2(2x)\right)
    Last edited by Archie Meade; March 8th 2010 at 12:53 PM. Reason: typo
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  4. #4
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    An alternate way

    You can try integration by parts on the following:

    \int(sin^4xcosx)cosxdx

    letting u = cosx and dv = sin^4xcosx dx
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