Question

The value of ${\int }_2^3\frac{\sqrt{x}}{\sqrt{5-x}+\sqrt{x}}dx$ is

• A. $1$
• B. $\frac{1}{2}$
• C. $2$
• D. None of these

Answer 1

Answer: (B)

$\frac{1}{2}$

Let $I={\int }_2^3\frac{\sqrt{x}}{\sqrt{5-x}+\sqrt{x}}dx$
$\therefore I={\int }_2^3\frac{\sqrt{5-x}}{\sqrt{x}+\sqrt{5-x}}dx$
On adding Eqs. (i) and (ii), we get
$2I={\int }_2^3\frac{\sqrt{x}+\sqrt{5-x}}{\sqrt{x}+\sqrt{5-x}}dx$
$={\int }_2^{3}1dx=3-2=1$
$\Rightarrow I=\frac{1}{2}$.