Question

Evaluate $\int 4\sin x\cos \frac{x}{2}\cos \frac{3x}{2}dx$

Answer 1

$\int 4\sin x\cos \frac{x}{2}\cos \frac{3x}{2}dx$

$=\int 2\sin x\left(2\cos \frac{x}{2}\cos \frac{3x}{2}\right)dx$

$=\int 2\sin x(\cos 2x+\cos x)dx$ using $2\cos A\cos B=\cos (A+B)+\cos (A-B)$

$=\int 2\sin x\cos 2xdx+\int 2\sin x\cos xdx$

$=\int 2\sin x\cos 2xdx+\int \sin 2xdx$

$=\int (\sin 3x+\sin (-x))dx+\int \sin 2xdx$

$=\int \sin 3xdx-\int \sin xdx+\int \sin 2xdx$

$=-\frac{\cos 3x}{3}+\cos x-\frac{\cos 2x}{2}+c$

$=-\frac{\cos 3x}{3}-\frac{\cos 2x}{2}+\cos x+c$