Question

Find the integral of the function: $\frac{\cos x}{1+\cos x}$

Answer 1

To find: $\int \frac{\cos x}{1+\cos x}dx$

Add and subtract by $1$ in numerator

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

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

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

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

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

$[\because 1+\cos x=2{\cos }^2\frac{x}{2}]$

$=\int (1-\frac{1}{2}{\sec }^2\frac{x}{2})dx$

$=x-\frac{1}{2}\left[\frac{\tan \frac{x}{2}}{\frac{1}{2}}\right]+C$

$=x-\tan \frac{x}{2}+C$

Where $C$ is constant of integration.