Question

Evaluate $\int \frac{dx}{(\sin \frac{x}{2}+\cos \frac{x}{2}{)}^2}$

Answer 1

$\int \frac{1}{{(\sin \frac{x}{2}+\cos \frac{x}{2})}^2}dx=\int \frac{1}{{\sin }^2\frac{x}{2}+{\cos }^2\frac{x}{2}+2\sin \frac{x}{2}\cos \frac{x}{2}}dx=\int \frac{1}{1+\sin x}dx=\int \frac{1-\sin x}{1-{\sin }^{2}x}dx=\int \frac{1-\sin x}{{\cos }^{2}x}dx=\int {\sec }^{2}x-\tan x\sec xdx=\tan x-\log \left|\tan x\right|+c$