Question

Evaluate: $\int \frac{1}{1+tanx}dx$

Answer 1

$l\int \frac{dx}{1+\tan x}$

$=\int \frac{\cos xdx}{\cos x+\sin x}$

$=\frac{1}{2}\int \frac{\left(\sin x+\cos x\right)+\left(\cos x-\sin x\right)}{\left(\sin x+\cos x\right)}dx$

$=\frac{x}{2}+\frac{1}{2}\int \frac{\left(\cos x-\sin x\right)}{\sin x+\cos x}dx$

$\sin x+\cos x=t$

$=\frac{x}{2}+\frac{1}{2}\int \frac{dt}{t}$

$I=\frac{x}{2}+\frac{1}{2}\log \left(\sin x+\cos x\right)+C$