Question

${\int }_0^a\frac{\sqrt{x}}{\sqrt{x}+\sqrt{a-x}}dx$

Answer 1

We have,

$I={\int }_0^a\frac{\sqrt{x}}{\sqrt{x}+\sqrt{a-x}}dx$ $........\left(1\right)$

Apply property,

$I={\int }_0^a\frac{\sqrt{a-x}dx}{\sqrt{x}+\sqrt{a-x}}$ $........\left(2\right)$

On adding equation (1) and (2), we get

$2I={\int }_0^{a}dx=aI=\frac{a}{2}$

Hence, this is the answer.