$\int \sqrt{1+2\cot x(\cos ecx+\cot x)}dx\left(0<x<\frac{\pi }{2}\right)$ is equal to (where C is a constant of integration) 

$\begin{array}{rcl} & & \int \sqrt{1+2\mathrm{cotx}(\mathrm{cosecx}+\mathrm{cotx})}\mathrm{dx}\\ & =& \int \sqrt{1+2\cot \mathrm{x} \mathrm{cosecx}+2{\cot }^2\mathrm{x}}\mathrm{dx}\\ & =& \int \sqrt{{\mathrm{cosec}}^2\mathrm{x}+2\mathrm{cotx} \mathrm{coxecx}+{\cot }^2\mathrm{x}}\mathrm{dx}\\ & =& \int \sqrt{{(\mathrm{cosec} \mathrm{x}+\cot \mathrm{x})}^2}\mathrm{dx}\\ & =& \int \left(\mathrm{cosec} \mathrm{x}+\cot \mathrm{x}\right)\mathrm{dx} \left(\because 0<\mathrm{x}<\frac{\mathrm{\pi }}{2}\right)\\ & =& \int \frac{1+\cos \mathrm{x}}{\sin \mathrm{x}}\mathrm{dx}\\ & =& \int \cot \frac{\mathrm{x}}{2}\mathrm{dx}\\ & =& 2 \log \sin \frac{\mathrm{x}}{2}+\mathrm{C}\\ & & \end{array}$