Question

Integrate the function:$\frac{1}{x+x\log x}$

Answer 1

$I=\frac{1}{x+x\log x}$

$=\frac{1}{x(1+\log x)}$

Let $1+\log x=t$

$\Rightarrow \frac{1}{x}dx=dt$

$\Rightarrow dx=x.dt$

$\int \frac{dx}{x(1+\log x)}$

$=\int \frac{x.dt}{x\left(t\right)}$

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

$=\log \left|t\right|+c$

$=\log \left|(1+\log x)\right|+c$ where $t=(1+\log x)$