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Math Help - Integrate sin^n(x)cos^m(x)

  1. #1
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    Integrate sin^n(x)cos^m(x)

    I need to integrate I=\int\sin^n(x)\cos^m(x)\,dx where (n,m)\in \mathbb{R}. How do I do that?
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  2. #2
    MHF Contributor Also sprach Zarathustra's Avatar
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    Here are some ideas for start...

    Integrals: sin^n(x)cos^m(x), sinh^n(x)cosh^m(x)
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    The problem I'm having is \exists (m,n)\notin\mathbb{N} so no integration by parts.
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  4. #4
    MHF Contributor Also sprach Zarathustra's Avatar
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    But, \mathbb{N}\subset \mathbb{R}, anyway I hardly believe you find such formula, you can find close formula for m,n\in \mathbb{Z}, I think it is the top, but maybe I wrong here...
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    For any [LaTeX ERROR: Convert failed] , the derivative [LaTeX ERROR: Convert failed] . Similarly for the antiderivative. In the integration by parts you are using the above plus the chain rule so the integration should work for [LaTeX ERROR: Convert failed] . Of course, you are going to have to exclude some special values like (n,m) = (0,0) or (1,1), etc

    Although if you try n = 2/3 and m = 3/4, Mathematica will give you something involving hypergeometric series.
    If you try n = 1/2 and m = 2, you get an elliptic integral.

    I did some numerical integration for values of n and m for the integration by parts identity and it all works well as long as we work in the given ranges, i.e. n, m >1.
    Last edited by Vlasev; July 29th 2010 at 05:07 PM.
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  6. #6
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    Mathematica gives me \int\sin^mx\cos^nx\,dx=-\,_2F_1\left(\dfrac{1-m}{2},\dfrac{1+n}{2},\dfrac{3+n}{2},\cos^2x\right)  \dfrac{\sin^{1+m}x}{\left(\sqrt{\sin^2x}\right)^{1  +m}}\dfrac{1}{1+n}\cos^{1+n}x
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