Question

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

Answer 1

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

$=\frac{1}{2}\int \frac{2\cos x}{\cos x+\sin x}dx$

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

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

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

$=\frac{1}{2}\log |\cos x+\sin x|+\frac{1}{2}x+c$

Formula:

$\int \frac{f^{\prime }\left(x\right)}{f\left(x\right)}dx=\log f\left(x\right)+c$