Question

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

Answer 1

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

Let $\frac{\sin x}{3\cos x+2}=t$

$\Rightarrow \frac{(3\cos x+2)\cos x-\sin x(-3\sin x)}{{(3\cos x+2)}^2}dx=dt$

$\Rightarrow \frac{3+2\cos x}{{(3\cos x+2)}^{2}dx}=dt$

$\Rightarrow I=\int 1.dt=t+c=\frac{\sin x}{3\cos x+2}+c$