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

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

$\begin{array}{l}\text{ Multiplying }{\mathrm{N}}^{\mathrm{r}}\text{ and }{\mathrm{D}}^{\mathrm{r}}\text{ by }\cos e{c}^{2}x,\text{ we get }\\ I=\int \frac{\left(3{\mathrm{cosec}}^{2}x+2\cot x\mathrm{cosec}x\right)}{(2\cos ecx+3\cot x{)}^2}dx\\ =-\int \frac{-3{\mathrm{cosec}}^{2}x-2\cot x\mathrm{cosec}x}{(2\cos \mathrm{ex}x+3\cot x{)}^2}dx\\ =\frac{1}{2\mathrm{cosec}x+3\cot x}+C=\left(\frac{\sin x}{2+3\cos x}\right)+C\end{array}$