Question

Find the integrals of the function:

${\displaystyle \frac{\cos x}{1+\cos x}}$

Answer 1

Let ${\displaystyle I=\int \frac{\cos x}{1+\cos x} dx}$

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

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

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

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

${\displaystyle =x=\frac{1}{2}\cdot \frac{\tan \frac{x}{2}}{\frac{1}{2}}+C}$

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