Question

The value of $\int \frac{3+2\cos x}{(2+3\cos x{)}^2}dx$ is equal to

• A. $\frac{\sin x}{3\cos x+2}+c$
• B. $\frac{2\cos x}{3\sin x+2}+c$
• C. $\frac{2\cos x}{3\cos x+2}+c$
• D. $\frac{2\sin x}{3\sin x+2}+c$

Answer 1

The correct option is A $\frac{\sin x}{3\cos x+2}+c$

Let $I=\int \frac{3+2\cos x}{{(2+3\cos x)}^2}dx$
Substitute $u=\tan \frac{x}{2}\Rightarrow du=\frac{1}{2}{\sec }^2\frac{x}{2}dx$
$\therefore \int \frac{2(u^2+5)}{u^4-10u^2+25}du=2\int \frac{u^2+5}{{(u^2-5)}^2}du$
$=\int (\frac{1}{2{(u+\sqrt{5})}^2}+\frac{1}{2{(\sqrt{5}-u)}^2})du$
$=-\frac{2u}{u^2-5}=\frac{\sin x}{3\cos x+2}$