1. ## 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?

2. Here are some ideas for start...

Integrals: sin^n(x)cos^m(x), sinh^n(x)cosh^m(x)

3. The problem I'm having is $\exists (m,n)\notin\mathbb{N}$ so no integration by parts.

4. 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...

5. 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.

6. Mathematica gives me $\int\sin^mx\cos^nx\,dx=-\,_2F_1\left(\dfrac{1-m}{2},\dfrac{1+n}{2},\dfrac{3+n}{2},\cos^2x\right) \dfrac{\sin^{1+m}x}{\left(\sqrt{\sin^2x}\right)^{1 +m}}\dfrac{1}{1+n}\cos^{1+n}x$