Question

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

• A. $-\sin 2x-\sin x+c$
• B. $-\frac{1}{2}\sin 2x+\sin x+C$
• C. $-\frac{1}{2}\sin 2x-\sin x+C$
• D. $\frac{1}{2}\sin 2x+\sin x+c$

Answer 1

The correct option is C $-\frac{1}{2}\sin 2x-\sin x+C$

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

$\int \left[\frac{(\cos 5x+\cos x)+(\cos 4x+\cos 2x)-(\cos x+\cos 2x)}{(1-2\cos 3x)}\right]dx$

$\int \left[\frac{2\cos 3x\cos 2x+2\cos 3x\cos x-(\cos x+\cos 2x)}{1-2\cos 3x}\right]dx$

$\int \frac{(2\cos 3x-1)(\cos x+\cos 2x)}{1-2\cos 3x}dx$

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

$=-\sin x-\frac{1}{2}\sin 2xc$