Question

Evaluate :

$\int \frac{\cos 5x+\cos 4x}{1-2\cos 3x}$ dx

Answer 1

$I=\int \frac{\cos 5x+\cos 4x}{1-2\cos 3x}dx$

$I=\int \frac{\sin 3x(\cos 5x+\cos 4x)}{\sin 3x-2\sin 3x\cos 3x}dx$

$I=\int \frac{\left(2\sin \frac{3x}{2}\cos \frac{3x}{2}\right)\left(2cos\frac{9x}{2}\cos \frac{x}{2}\right)}{\sin 3x-\sin 6x}dx$

$I=4\int \frac{\sin \frac{3x}{2}\cos \frac{3x}{2}\cos \frac{9x}{2}\cos \frac{x}{2}}{-2\cos \frac{9x}{2}\sin \frac{3x}{2}}dx$

$I=-\int 2\cos \frac{3x}{2}\cos \frac{x}{2}dx$

$I=-\int (\cos 2x+\cos x)dx$

$I=-(\frac{\sin 2x}{2}+\sin x)+C$