$\begin{align}  & \int{\sin 7x\sin 5xdx}=\frac{1}{2}\int{[\cos (7x-5x)-\cos (7x+5x)]dx} \\ & \int{\sin 7x\sin 5xdx}\text{   =}\frac{1}{2}\int{(\cos 2x-\cos 12x)dx} \\ & \text{ }\int{\sin 7x\sin 5xdx}\text{ }=\frac{1}{2}\left[ \frac{\sin 2x}{2}-\frac{\sin 12x}{12} \right]+C \\\end{align}$