Chapter 8.3, Problem 68E

Integrals of the form ${\displaystyle \int \sin mx\cos nxdx}$ Use the following three identities to evaluate the given integrals.

$\begin{array}{lll}\sin mx\sin nx\hfill & =\hfill & \frac{1}{2}\left(\cos ((m-n)x)-\cos ((m+n)x)\right)\hfill \\ \sin mx\cos nx\hfill & =\hfill & \frac{1}{2}\left(\sin ((m-n)x)+\sin ((m+n)x)\right)\hfill \\ \cos mx\cos nx\hfill & =\hfill & \frac{1}{2}\left(\cos ((m-n)x)+\cos ((m+n)x)\right)\hfill \end{array}$

68. ${\displaystyle \int \sin 5x\sin 7xdx}$