Question

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

Answer 1

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

$\cos 5x+\cos 4x=2\cos \frac{9x}{2}\cos \frac{x}{2}$

$1-2\cos 3x=1-2(2{\cos }^2\frac{3x}{2}-1)$

$=3-4{\cos }^2\frac{3x}{2}$

$\frac{2\cos \frac{9x}{2}.\cos \frac{x}{2}}{3-4{\cos }^2\frac{3x}{2}}\times \frac{\cos 3\frac{x}{2}}{\cos 3\frac{x}{2}}$

$=\frac{2\cos \frac{x}{2}.\cos \frac{3x}{2}.\cos \frac{9x}{2}}{-(4\cos \frac{3x}{2}-3\cos \frac{3x}{2})}$

By using $\cos 3\theta =4{\cos }^3\theta -3\cos \theta$

$=\frac{2\cos \frac{x}{2}\cos \frac{3x}{2}.\cos \frac{9x}{2}}{-\cos \frac{9x}{2}}$

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

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

Now $I=\int -\cos 2x-\cos x.dx.$

$=\frac{-\sin 2x}{2}-\sin x+c$.