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

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

Then:


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

0 & : m \ne n \\ \dfrac \pi 2 & : m = n \end{cases}$

That is:
 * $\ds \int_0^\pi \sin m x \sin n x \rd x = \dfrac \pi 2 \delta_{m n}$

where $\delta_{m n}$ is the Kronecker delta.

Proof
Let $m \ne n$.

When $m = n$ we have: