Question

Evaluate : $\int \frac{3+2\cos x}{(2+3\cos x{)}^2}dx$

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

Answer 1

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

Let $I=\int \frac{3+2\cos x}{(2+3\cos x{)}^2}dx$.

Dividing $N^r$ and $D^r$ by $si{n}^{2}x$, we get

$I=\int \frac{(3{cosec}^{2}x+2\cot xcosecx)}{(2cosecx+3\cot x{)}^2}dx$

Put $2cosecx+3\cot x=t$

$\Rightarrow (-2\cot xcosecx-3{cosec}^{2}x)dx=dt$

$=-\int \frac{dt}{t^2}=-\frac{-1}{t}+C$

$=\frac{1}{2cosecx+3\cot x}+C=\left(\frac{\sin x}{2+3\cos x}\right)+C$