I need to integrate $\displaystyle I=\int\sin^n(x)\cos^m(x)\,dx$ where $\displaystyle (n,m)\in \mathbb{R}$. How do I do that?
Here are some ideas for start...
Integrals: sin^n(x)cos^m(x), sinh^n(x)cosh^m(x)
For any $\displaystyle n \in \mathbb{R}$, the derivative $\displaystyle \frac{d}{dx}x^n = n x^{n-1}$. Similarly for the antiderivative. In the integration by parts you are using the above plus the chain rule so the integration should work for $\displaystyle (n,m) \in \mathbb{R}$. 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.