Kindly consider the following

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

This is a Short Answer Type Questions as classified in NCERT Exemplar

Sol:

$\begin{array}{l}LetI={\displaystyle \int \frac{dx}{1+cosx}}\\ ={\displaystyle \int \frac{dx}{2co{s}^{2}x/2}\left[?1+cosx=2co{s}^{2}x/2\right]}\\ =\frac{1}{2}{\displaystyle \int se{c}^2\frac{x}{2}dx}=\frac{1}{2}.2tan\frac{x}{2}+C=tan\frac{x}{2}+C\\ Hence,therequiredsolutionistan\frac{x}{2}+C.\end{array}$