Question

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

• A. $\log \left[\cot x\tan \left(x/2\right)\right]$
• B. $\log \left[\tan x\tan \left(2x\right)\right]$
• C. $\log \left[\tan x\tan \left(x/2\right)\right]$
• D. $\log \left[\tan x\cot \left(x/2\right)\right]$

Answer 1

Answer: (C)

$\log \left[\tan x\tan \left(x/2\right)\right]$

$\int \frac{1+\cos x}{\sin x\cos x}dx=\int \frac{1}{\sin x\cos x}dx+\int \frac{\cos x}{\sin x\cos x}dx$
$=\int \frac{2}{\sin 2x}dx+\int \frac{1}{\sin x}dx$
$=2\int \text{cosec}2xdx+\int \text{cosec}xdx$
$=\log \tan x+\log \tan \left(x/2\right)=\log \left[\tan x\tan \left(x/2\right)\right]$

Hence, option 'C' is correct.