Question

$\underset{0}{\overset{a}{\int }}\frac{\sqrt{x}}{\sqrt{x}+\sqrt{a-x}}dx=$

• A. $a$
• B. $\frac{a}{2}$
• C. $2a$
• D. $\frac{a}{4}$

Answer 1

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

$I=\underset{0}{\overset{a}{\int }}\frac{\sqrt{x}}{\sqrt{x}+\sqrt{a-x}}dx$ ......(1)

$I=\underset{0}{\overset{a}{\int }}\frac{\sqrt{a-x}}{\sqrt{x}+\sqrt{a-x}}dx$ ........(2) $\left(\underset{0}{\overset{a}{\int }}f\right(x)dx=\underset{0}{\overset{a}{\int }}f(a-x\left)dx\right)$

Adding 1 and 2, we get,

$2I=\underset{0}{\overset{a}{\int }}dx$

$2I=a$

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