Question

Solve the following integral:

$\int x\cdot \sin 2x\cdot \sin 5xdx$.

Answer 1

Given integral is $\int x.\sin 2x.\sin 5xdx$,

We know that $\cos (A-B)-\cos (A+B)=2\sin A.\sin B$,

$⟹\int x.\sin 2x.\sin 5xdx=\frac{1}{2}[\int x.\cos 3xdx-\int x.\cos 7xdx]$,

We know that from integration by parts,

$\int x.\cos xdx=x.\sin x-\int \sin xdx=x.\sin x+\cos x+c$,

$=\frac{1}{2}[\frac{x.\sin 3x}{3}+\frac{\sin 3x}{3}-[\frac{x.\sin 7x}{7}+\frac{\sin 7x}{7}]]+c=\frac{x.\sin 3x}{6}+\frac{\sin 3x}{6}-\frac{x.\sin 7x}{14}-\frac{\sin 7x}{14}+c$