Question

Solve:
$\int \frac{1}{x+x\ln x}dx.$

Answer 1

Consider the given integral.

$I=\int \frac{xdx}{1+x\tan x}$

$I=\int \frac{xdx}{1+x\frac{\sin x}{\cos x}}$

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

Let $t=x\sin x+\cos x$

$\frac{dt}{dx}=x\cos x+\sin x-\sin x$

$dt=x\cos xdx$

Therefore,

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

$I=\ln \left(t\right)+C$

On putting the value of $t$, we get

$I=\ln (x\sin x+\cos x)+C$

Hence, this is the answer.