Question

Integrate:

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

Answer 1

$\int \frac{1}{1+cosxdx}=\int \frac{1-cosx}{(1+cosx)(1-cosx)}dx$

$=\int \frac{1-cosx}{1-cos2xdx}$

$=\int \frac{1-cosx}{{sin}^{2}xdx}$

$=\int \frac{1}{{sin}^{2}x}dx-\int \frac{cosx}{{sin}^{2}x}dx$

$=\int {cosec}^{2}xdx-\int cotxcosecxdx$

$=-cotx+cscx+C$