I often need to solve a type of integral which is :
$$\int x \ \sin ( n_1 x) \ \sin (n_2 x) \ dx $$ $n_1$ and $n_2$ are integers.
where the $n_1 $ and $n_2$ are the integer number. The way I solve this integral is to break down the $\sin n_1 x \sin n_2 x $ first using formula $\cos C - \cos D$ and then take the partial integration.
Is there any easy way to solve this integral?