Question

The integral $\int \sqrt{1+2\cot x(\text{cosec}x+\cot x)}dx$ $(0<x<\frac{\pi }{2})$ is equal to. (where C is a constant of integration).

• A. $2\log \left(\cos \frac{x}{2}\right)+C$
• B. $4\log \left(\sin \frac{x}{2}\right)+C$
• C. $4\log \left(\cos \frac{x}{2}\right)+C$
• D. $2\log \left(\sin \frac{x}{2}\right)+C$

Answer 1

The correct option is D $2\log \left(\sin \frac{x}{2}\right)+C$

$\int \left(\sqrt{2\cot x\text{cosec}x+{\text{cosec}}^{2}x+{\cot }^{2}x}\right)dx$

$\Rightarrow \int |\text{cosec}x+\cot x|dx$

$\Rightarrow \int (\text{cosec}x+\cot x)dx$

$\Rightarrow \int \frac{1+\cos x}{\sin x}dx$

$\Rightarrow \int \cot \frac{x}{2}dx$

$\Rightarrow 2\log \left(\sin \frac{x}{2}\right)+c$