Question

Evaluate the following indefinite integrals:
cos x sin 5x dx

Answer 1

Angle sum/difference identity for sine:
\[(1)~~\sin(x+y)=\sin x\cos y+\sin y\cos x\\
(2)~~\sin(x-y)=\sin x\cos y-\sin y\cos x\]
Observe that adding the two together gives the following:
\[\sin(x+y)+\sin(x-y)=2\sin x\cos y\]
\[\frac{\sin(x+y)+\sin(x-y)}{2}=\sin x\cos y\]
So for this integral, rewrite by replacing \(x\) with \(5x\) and \(y\) with \(x\):
\[\int\cos x\sin5x~dx=\frac{1}{2}\int\left(\sin6x+\sin4x\right)~dx\]