Question

Evaluate:
$\int \frac{1}{x-\sqrt{x}}dx$

Answer 1

$I=\int \frac{1}{x-\sqrt{x}}dx$

$x=u^2$

$\Rightarrow dx=2udu$

$\Rightarrow I=\int \frac{1}{u^2-u}2udu$

$=2\int \frac{du}{u-1}$

$=2ln(|u-1|)+c$

$=2ln(|\sqrt{x}-1|)+c$.