By using the properties of definite integrals, evaluate the integrals `int_0^a(sqrt(x))/(sqrt(x)+sqrt(a-x))dx`

Explanation

Given:`int_0^a(sqrt(x))/(sqrt(x)+sqrt(a-x))dx`

`I=int_0^a(sqrt(x))/(sqrt(x)+sqrt(a-x))dx......(i)`

As we know ` int_0^a f(x)dx=int_0^a f(a-x)dx`

`I=int_0^a(sqrt(a-x))/(sqrt(a-x)+sqrt(x))dx.......(ii)`

Now add (i) and (ii) we get

`2I=int_0^a(sqrt(x)+sqrt(a-x))/(sqrt(x)+sqrt(a-x))dx`

`2I=int_0^a 1dx`

`2I=[x]_0^a`

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