Question

Evaluate : $\int \frac{x}{1+x\tan x}dx$

Answer 1

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

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

Let $\cos x+\sin x=t⟹dt=x\cos xdx$

$=\int \frac{1}{t}dt$

$=\ln |t|+c$

$=\ln |\cos x+x\sin x|$