Question

$\underset{0}{\overset{a}{\int }}\frac{\sqrt{x}}{\sqrt{x}-\sqrt{a-x}}dx$ is eqal to

• A. $a$
• B. $\frac{a}{2}$
• C. $\sqrt{\frac{a-x}{a}}$
• D. $0$

Answer 1

The correct option is B $\frac{a}{2}$

Let $I=\underset{0}{\overset{a}{\int }}\frac{\sqrt{x}}{\sqrt{x}-\sqrt{a-x}}dx...\left(i\right)$
using property of integration
$I=\underset{0}{\overset{a}{\int }}\frac{\sqrt{a-x}}{\sqrt{a-x}-\sqrt{x}}dx=-\underset{0}{\overset{a}{\int }}\frac{\sqrt{a-x}}{\sqrt{x}-\sqrt{a-x}}dx...\left(ii\right)$
Adding equation $\left(i\right)$ and $\left(ii\right)$ we get,
$2I=\underset{0}{\overset{a}{\int }}\frac{\sqrt{x}-\sqrt{a-x}}{\sqrt{x}-\sqrt{a-x}}dx=\underset{0}{\overset{a}{\int }}1\cdot dx=(a-0)=a$
$I=\frac{a}{2}$