Integral over 2 pi of Sine of m x by Sine 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 \sin n x \, \mathrm d x = \begin{cases}

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

That is:
 * $\displaystyle \int_\alpha^{\alpha + 2 \pi} \sin m x \sin n x \, \mathrm d x = \pi \delta_{m n}$

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

Proof
Let $m \ne n$.

When $m = n$ we have: