Integral over 2 pi of Sine of m x by Cosine of n x

Theorem
Let $m, n \in \Z$ be integers.

Let $\alpha \in \R$ be a real number.

Then:


 * $\displaystyle \int_\alpha^{\alpha + 2 \pi} \sin m x \cos n x \rd x = 0$

Proof
Let $m \ne n$.

When $m = n$ we have: