Question

Integrate the function: $\frac{1}{x-\sqrt{x}}$

Answer 1

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

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

Let $\sqrt{x}-1=t$

$\Rightarrow \frac{1}{2\sqrt{x}}dx=dt$ ( On differentiating w.r.t $x$ )

$=\int \frac{2dt}{t}$

$=2\log t+c$

$=2\log (\sqrt{x}-1)+c$

Hence, the answer is $2\log (\sqrt{x}-1)+c.$