$\int \frac{3+2 \cos x}{(2+3 \cos x)^2} d x$ is equal to

$\left(\frac{\sin x}{3 \cos x+2}\right)+c$

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

Multiplying Nr. & Dr. by cosec²x

$\Rightarrow I=\int \frac{\left(3 ~cosec^2 x+2 \cot x ~cosec ~x\right)}{(2 cosec ~x+3 \cot x)^2} dx$

$=-\int \frac{-3 cosec^2x - 2 \cot x ~cosec ~x}{(2 ~cosec ~x+3 \cot x)^2} dx$

$=\frac{1}{2 cosec ~x+3 \cot x}=\left(\frac{\sin x}{2+3 \cos x}\right)+c$

Hence (1) is the correct answer.