Question

Integral of $\sqrt{1+2\cot x(\cot x+cosecx)}$ w.r.t x is :

• A. $2\ln \cos \frac{x}{2}+c$
• B. $2\ln \sin \frac{x}{2}+c$
• C. $\frac{1}{2}\ln \cos \frac{x}{2}+c$
• D. $\ln \sin x-\ln (cosecx-\cot x)+c$

Answer 1

The correct option is B $2\ln \sin \frac{x}{2}+c$

Given , $\sqrt{1+2\cot x(\cot x+cosecx)}$
$\Rightarrow$ $\sqrt{1+2{\cot }^{2}x+2\cot xcosecx}$
$\Rightarrow$ $\sqrt{1+2\frac{{\cos }^{2}x}{{\sin }^{2}x}+2\frac{\cos x}{{\sin }^{2}x}}$
$\Rightarrow$ $\sqrt{\frac{{\sin }^{2}x}{{\sin }^{2}x}+2\frac{{\cos }^{2}x}{{\sin }^{2}x}+2\frac{\cos x}{{\sin }^{2}x}}$
$\Rightarrow$ $\frac{\sqrt{{\sin }^{2}x+2{\cos }^{2}x+2\cos x}}{\sin x}$
$\Rightarrow$ $\frac{\sqrt{1-{\cos }^{2}x+2{\cos }^{2}x+2\cos x}}{\sin x}$
$\Rightarrow$ $\frac{\sqrt{{\cos }^{2}x+2\cos x+1}}{\sin x}$
$\Rightarrow$ $\frac{\sqrt{(\cos x+1{)}^2}}{\sin x}$
$\Rightarrow$ $\frac{(\cos x+1)}{\sin x}$
writing $\sin x$ and $\cos x$ in submultiple angle we have,
$\Rightarrow$ $\frac{(2{\cos }^2\frac{x}{2}-1+1)}{2\sin \frac{x}{2}\cos \frac{x}{2}}$
$\Rightarrow$ $\frac{\left(2\cos \frac{x}{2}\right)}{2\sin \frac{x}{2}}$
$\Rightarrow$ $\cot \frac{x}{2}$
Since , $\int \cot x=ln|\sin x|+c$
So, $\int \cot \frac{x}{2}=2ln|\sin \frac{x}{2}|+c$
Hence, option 'B' is correct.