Question

Evaluate

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

Answer 1

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

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

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

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

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

$=x-\frac{1}{4}\tan \frac{x}{2}+c$