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

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

….(i)
$\Rightarrow$ $I = \int\limits_0^a {\cfrac{{\sqrt {a - x} }}{{\sqrt {a - x} + \sqrt {a - \left( {a - x} \right)} }}dx} = \int\limits_0^a {\cfrac{{\sqrt {a - x} }}{{\sqrt {a - x} + \sqrt x }}dx}$

….(ii)
Adding (i) and (ii),

we get
$2I = \int\limits_0^a {\cfrac{{\sqrt x + \sqrt {a - x} }}{{\sqrt x + \sqrt {a - x} }}dx = \int\limits_0^a {\left( 1 \right)dx} = \left[ x \right]_0^a} = a - 0 = a$

$\therefore$ $I = \cfrac{a}{2}$