Question

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

Answer 1

$I=\int \left(\frac{dx}{x-\sqrt{x}}\right)$

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

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

$x=t^2,dx=2tdt$

$I=\int \frac{2tdt}{t-1}-\frac{2t}{t}dt$

$I=\int \frac{2(t-1+1)dt}{t-1}-2dt$

$I=2t+2\ln (t-1)-2t$

$I=2\sqrt{x}+2\ln (\sqrt{x}-1)-2\sqrt{x}$

$I=2\ln (\sqrt{x}-1)$