Browse Questions

# Find the integrals of the functions$\frac{\cos x}{1+\cos x}$

$\begin{array}{1 1}x-\tan \frac{x}{2}+c \\ x+\tan \frac{x}{2}+c \\ x-\cot \frac{x}{2}+c \\ x+\cot \frac{x}{2}+c \end{array}$

Toolbox:
• $1+\cos x=2cos^2\frac{x}{2}.$
• $\cos x=2\cos^2\frac{x}{2}-1.$
• $\int\sec^2x=\tan x+c.$
Given:$I=\int \frac{\cos x}{1+\cos x}$

Using the information from the tool box we get,

$\cos x=2\cos^2\frac{x}{2}-1$ and $1+\cos x=2\cos^2\frac{x}{2}.$

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

Now seperating the terms we get,

$I=\int\big(1-\frac{1}{2\cos^2\frac{x}{2}}\big)dx.$

separating the terms we get,

$\;\;\;=\int dx-\frac{1}{2}\int \sec^2\frac{x}{2}dx.$

On integrating we get,

$\;\;\;=x-\frac{1}{2}\frac{\tan\frac{x}{2}}{\frac{1}{2}}+c.$

$\;\;\;=x-\tan x/2+c.$