I need to integrate where . How do I do that?

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- July 29th 2010, 12:06 PMfobos3Integrate sin^n(x)cos^m(x)
I need to integrate where . How do I do that?

- July 29th 2010, 12:19 PMAlso sprach Zarathustra
Here are some ideas for start...

Integrals: sin^n(x)cos^m(x), sinh^n(x)cosh^m(x) - July 29th 2010, 04:03 PMfobos3
The problem I'm having is so no integration by parts.

- July 29th 2010, 04:20 PMAlso sprach Zarathustra
But, , anyway I hardly believe you find such formula, you can find close formula for , I think it is the top, but maybe I wrong here...

- July 29th 2010, 04:31 PMVlasev
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. - July 30th 2010, 03:12 AMfobos3
Mathematica gives me