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

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

$2I=[x]_0^a$

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

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Subjective Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 86

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