Question

\[\int\limits_0^a \frac{\sqrt{x}}{\sqrt{x} + \sqrt{a - x}} dx\]

Answer 1

\[Let I = \int_0^a \frac{\sqrt{x}}{\sqrt{x} + \sqrt{a - x}} d x .............(1)\]

\[ \Rightarrow I = \int_0^a \frac{\sqrt{a - x}}{\sqrt{a - x} + \sqrt{x}}dx .....................\left[\text{Using, }\int_0^a f\left( x \right) dx = \int_0^a f\left( a - x \right) dx \right]\]
\[ \Rightarrow I = \int_0^a \frac{\sqrt{a - x}}{\sqrt{x} + \sqrt{a - x}}dx .......................(2)\]
\[\text{Adding (1) and } (2)\]
\[2I = \int_0^a \frac{\sqrt{x} + \sqrt{a - x}}{\sqrt{x} + \sqrt{a - x}} dx\]
\[ = \int_0^a dx = \left[ x \right]_0^a = a\]
\[Hence\ I = \frac{a}{2}\]