# How do you find the integral of int 1/(1 + cos(x))?

Sep 18, 2015

$- \cot x + \csc x + \text{C}$

#### Explanation:

$\int \frac{1}{1 + \cos x} \mathrm{dx} = \int \frac{1 - \cos x}{\left(1 + \cos x\right) \left(1 - \cos x\right)} \mathrm{dx}$

$= \int \frac{1 - \cos x}{1 - {\cos}^{2} x} \mathrm{dx}$

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

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

=$\int {\csc}^{2} x \mathrm{dx} - \int \cot x \csc x \mathrm{dx}$

=$- \cot x + \csc x + \text{C}$