Question

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

Answer 1

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

As we know that $\cos 2\theta =1-2{\sin }^2\theta \Rightarrow 1+\cos 2\theta =2{\sin }^2\theta$

$\therefore \int \frac{1}{1+\cos x}dx=\int \frac{1}{2{\sin }^2\frac{x}{2}}dx$

$\Rightarrow =\frac{1}{2}\int {\csc }^2\frac{x}{2}dx$

$\Rightarrow =\frac{1}{2}(-2\cot \frac{x}{2}+C)$

$\Rightarrow =-\cot \frac{x}{2}+C$

Hence $\int \frac{1}{1+\cos x}dx=-\cot \frac{x}{2}+C$