Question

Evaluate $\int \frac{1}{1+\sin x+\cos x}dx$.

Answer 1

Let $tan\left(\frac{x}{2}\right)=t$ , then $\frac{1}{2}{\sec }^2\left(\frac{x}{2}\right)dx=dt$

So we get $dx=\left(\frac{2}{1+t^2}\right)dt$

$sinx=\frac{2t}{1+t^2},cosx=\frac{1-t^2}{1+t^2}$

Therefore given integral will transform into $\int \frac{1}{1+\frac{2t}{1+t^2}+\frac{1-t^2}{1+t^2}}\times \frac{2}{1+t^2}dt=\int \frac{2}{2+2t}dt=\int \frac{1}{1+t}dt=\ln (1+t)=\ln (1+\tan \left(\frac{x}{2}\right))$