$\int\limits_0^1 {\cfrac{{dx}}{{\sqrt {1 + x} - \sqrt x }}}$

Let $I = \int\limits_0^1 {\cfrac{{dx}}{{\sqrt {1 + x} - \sqrt x }}} = \int\limits_0^1 {\cfrac{{\sqrt {1 + x} + \sqrt x }}{{1 + x - x}}} dx$

$= \int\limits_0^1 {\left[ {\sqrt {1 + x} + \sqrt x } \right]dx} = \left[ {\cfrac{2}{3}{{\left( {1 + x} \right)}^{3/2}}} \right]_0^1 + \left[ {\cfrac{2}{3}{x^{3/2}}} \right]_0^1$

$= \cfrac{2}{3}\left( {{2^{3/2}} - 1} \right) + \cfrac{2}{3} = \cfrac{2}{3} \cdot {2^{3/2}} - \cfrac{2}{3} + \cfrac{2}{3} = \cfrac{2}{3} \cdot 2\sqrt 2 = \cfrac{{4\sqrt 2 }}{3}$