Question

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

Answer 1

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

\[ = \int_0^a \frac{\sqrt{a - x}}{\sqrt{a - x} + \sqrt{a - a + x}} d x\]

\[ = \int_0^a \frac{\sqrt{a - x}}{\sqrt{a - x} + \sqrt{x}} d x\]

\[ \Rightarrow I = \int_0^a \frac{\sqrt{a - x}}{\sqrt{x} + \sqrt{a - x}} d x.................(2)\]

Adding (1) and (2)

\[2I = \int_0^a \left[ \frac{\sqrt{x}}{\sqrt{x} + \sqrt{a - x}} + \frac{\sqrt{a - x}}{\sqrt{x} + \sqrt{a - x}} \right] d x\]

\[ = \int_0^a dx\]

\[ = \left[ x \right]_0^a \]

\[ = a\]

\[\text{Hence, }I = \frac{a}{2}\]