The integral ∫ 1 + 2 cot x cosec x + cot x d x ,   0 < x < π 2   is equal to

Let $I=\int \sqrt{1+2\cot x\left(\mathrm{cosec} x+\cot x\right)}dx$

$\Rightarrow I=\int \left(\sqrt{2\cot x\mathrm{cosec}x+{\mathrm{cosec}}^{2}x+{\cot }^{2}x}\right)dx$

$\Rightarrow I=\int \sqrt{{\left(\mathrm{cosec} x+\cot x\right)}^2}dx$

$\Rightarrow I=\int \left|\mathrm{cosec} x+\cot x\right|dx$

$\Rightarrow I=\int \left(\mathrm{cosec} x+\cot x\right)dx$

$\Rightarrow I=\log \left|\tan \frac{x}{2}\right|+\log \left|\sin x\right|+c$, where $c$ is the constant of integration.
$\Rightarrow I=\log \left(\frac{\sin {\displaystyle \frac{x}{2}}}{\cos {\displaystyle \frac{x}{2}}}·2\sin \frac{x}{2}\cos \frac{x}{2}\right)+c$
$\Rightarrow I=2\log \left|\sin \frac{x}{2}\right|+c$