Mathematics

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


SOLUTION
We have,
$$I=\int\limits_0^a {\dfrac{{\sqrt x }}{{\sqrt x  + \sqrt {a - x} }}dx} $$        $$........(1)$$

Apply property,
$$I=\int\limits_a^b f(x) dx=\int\limits_a^bf(a+b-x)dx$$

Therefore,
$$I=\int\limits_0^a {\dfrac{{\sqrt {a-x} }}{{\sqrt {a-x}  + \sqrt {a-a + x} }}dx} $$

$$I=\int\limits_0^a {\dfrac{{\sqrt {a-x} }}{{\sqrt {a-x}  + \sqrt { x} }}dx} $$          $$.......(2)$$

On adding both equations, we get
$$2I=\int\limits_0^a {\dfrac{{\sqrt {a-x} +\sqrt x}}{{\sqrt {a-x}  + \sqrt {x} }}dx} $$

$$2I=\int\limits_0^a 1dx $$

$$2I=[x]_0^a $$

$$I=\dfrac{a}{2} $$

Hence, this is the answer.
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Subjective Medium Published on 17th 09, 2020
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