∫limits_-π^π sin m x sin n x d x (m ≠ n and m, n are integers) =

$I=\frac{1}{2} \int\limits_0^\pi 2 \sin m x \cos n x d x$

$=\frac{1}{2} \int\limits_0^\pi \cos (m-n) x-\cos (m+n) x d x$

$=\frac{1}{2}\left[\frac{\sin (m-n) x}{m-n}-\frac{\sin (m+n) x}{m+n}\right]_0^\pi=0$