$\cfrac{1}{{x - \sqrt x }}$

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

Put $\sqrt x - 1 = t$ $\Rightarrow$ $\cfrac{1}{{2\sqrt x }}dx = dt$

$\therefore$ $I = 2\int {\cfrac{{dx}}{{\left( {\sqrt x - 1} \right)2\sqrt x }} = 2\int {\cfrac{{dt}}{t} = 2\log t} + C = 2\log \left( {\sqrt x - 1} \right) + C}$