Question

By using the properties of definite integrals, evaluate the integral: $\int_{o}^{a} \frac{\sqrt{x}}{\sqrt x+\sqrt{a-x} }dx$

Answer 1

Let I=$\int_{o}^{a} \frac{\sqrt{x}}{\sqrt x+\sqrt{a-x} }dx$………....(1)

It is known that,$\int_{o}^{a}f(x)dx=\int_{o}^{a}f(a-x)dx$

I=$\int_{o}^{a}\frac{\sqrt{a-x}}{\sqrt{a-x}+\sqrt x}dx$…………...(2)

Adding(1)and(2),we obtain

2i=$\int_{o}^{a}\frac{x+\sqrt{a-x}}{x+\sqrt{a-x}}dx$

⇒2I=$\int_{o}^{a}1.dx$

$⇒2I=[x]^a_0$

⇒2I=a

⇒I=$\frac{a}{2}$