Question

If $\int \frac{\cos 7x-\cos 8x}{1+2\cos 5x}$ $=$ $\frac{1}{k}\sin mx-\frac{1}{n}\sin px$, then $k+m+n+p$ is equal to

• A. 10
• B. 11
• C. 12
• D. 13

Answer 1

The correct option is A 10

$I=\int \frac{2\sin \frac{15x}{2}\sin \frac{x}{2}}{1+2\cos 5x}dx$
Multiply above and below by $\sin \frac{5x}{2}$
$\therefore D^r=\sin \frac{5x}{2}+2\cos 5x\sin \frac{5x}{2}$
$=\sin \frac{5x}{2}+\sin \frac{15x}{2}-\sin \frac{5x}{2}=\sin \frac{15x}{2}$
$\therefore \int \frac{2\sin \frac{15x}{2}\sin \frac{x}{2}\sin \frac{5x}{2}}{\sin \frac{15x}{2}}dx\int 2\sin \frac{x}{2}\sin \frac{5x}{2}dx$
$=\int (\cos 2x-\cos 3x)dx=\frac{1}{2}\sin 2x-\frac{1}{3}\sin 3x$
$\Rightarrow k+m+n+p=10$