Question

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

Answer 1

$\frac{\cos x}{1+\cos x}=\frac{{\cos }^2\frac{x}{2}-{\sin }^2\frac{x}{2}}{2{\cos }^2\frac{x}{2}}$
$=\frac{1}{2}[1-{\tan }^2\frac{x}{2}]$
$\therefore \int \frac{\cos x}{1+\cos x}dx=\frac{1}{2}\int (1-{\tan }^2\frac{x}{2})dx$
$=\frac{1}{2}\int (1-{\sec }^2\frac{x}{2}+1)dx$
$=\frac{1}{2}\int (2-{\sec }^2\frac{x}{2})dx$
$=\frac{1}{2}[2x-\frac{\tan \frac{x}{2}}{\frac{1}{2}}]+C$
$=x-\tan \frac{x}{2}+C$