Question

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

Answer 1

$\int \frac{dx}{x-\sqrt{x}}$

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

$=\int \frac{2dx}{2\sqrt{x}(\sqrt{x}-1)}$

Let $\sqrt{x}=t\Rightarrow \frac{1}{2\sqrt{x}}dx=dt$

$\therefore 2\int \frac{dt}{t-1}=2\log |t-1|$

$\because t=\sqrt{x}$

$\therefore \int \frac{dx}{x-\sqrt{x}}=2\log |\sqrt{x}-1|$