Question:
$\int \frac{d x}{1+\cos x}$
Solution:
Let $\mathrm{I}=\int \frac{d x}{1+\cos x}=\int \frac{d x}{2 \cos ^{2} x / 2} \quad\left[\because 1+\cos x=2 \cos ^{2} \frac{x}{2}\right]$
$=\frac{1}{2} \int \sec ^{2} \frac{x}{2} d x=\frac{1}{2} \cdot 2 \tan \frac{x}{2}+C=\tan \frac{x}{2}+C$
Therefore,
$I=\tan \frac{x}{2}+C$