Definite Integral from 0 to Pi of Sine of m x by Cosine of n x

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

Then:
 * $\ds \int_0^\pi \sin m x \cos n x \rd x = \begin{cases}

0 & : m + n \text { even} \\ \dfrac {2 m} {m^2 - n^2} & : m + n \text { odd} \end{cases}$

Proof
First we address the special case where $m = n$.

In this case $m + n = m + m = 2 m$ is even.

We have:

So in this case, $\ds \int_0^\pi \sin m x \cos n x \rd x = 0$ for $m + m$ even.

Let $m \ne n$.

When $m + n$ is an even integer, we have:

This shows that in all cases $\ds \int_0^\pi \sin m x \cos n x \rd x = 0$ for $m + m$ even.

When $m + n$ is an odd integer, we have: