JEE Main 2017 (Online) 8th AprilQuestion 1
The integral $\int\sqrt{1+2\cot x(\operatorname{cosec}x+\cot x)}\,dx$, $\left(0<x<\frac\pi2\right)$ is equal to :
Indefinite Integration
Solution
Concept. The expression under the root is $(\csc x+\cot x)^2$, and $\csc x+\cot x=\cot\frac x2$. Formulas used. (1) $1+\cot^2x=\csc^2x$(2) $\csc x+\cot x=\dfrac{1+\cos x}{\sin x}=\cot\dfrac x2$Step 1 - Perfect square. By (1), $$1+2\cot x\csc x+2\cot^2x=\csc^2x+2\cot x\csc x+\cot^2x=(\csc x+\cot x)^2.$$ Step 2 - Simplify. On $\left(0,\frac\pi2\right)$, $\csc x+\cot x>0$, so by (2) the integrand is $\cot\frac x2$. Step 3 - Integrate. $$\int\cot\frac x2\,dx=2\ln\left|\sin\frac x2\right|+c.$$ Answer: (A)