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Question
\[\int\limits_0^a \frac{\sqrt{x}}{\sqrt{x} + \sqrt{a - x}} dx\]
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Solution
\[Let, I = \int_0^a \frac{\sqrt{x}}{\sqrt{x} + \sqrt{a - x}} d x ...............(1)\]
\[ = \int_0^a \frac{\sqrt{a - x}}{\sqrt{a - x} + \sqrt{a - a + x}} d x\]
\[ = \int_0^a \frac{\sqrt{a - x}}{\sqrt{a - x} + \sqrt{x}} d x\]
\[ \Rightarrow I = \int_0^a \frac{\sqrt{a - x}}{\sqrt{x} + \sqrt{a - x}} d x.................(2)\]
Adding (1) and (2)
\[2I = \int_0^a \left[ \frac{\sqrt{x}}{\sqrt{x} + \sqrt{a - x}} + \frac{\sqrt{a - x}}{\sqrt{x} + \sqrt{a - x}} \right] d x\]
\[ = \int_0^a dx\]
\[ = \left[ x \right]_0^a \]
\[ = a\]
\[\text{Hence, }I = \frac{a}{2}\]
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