Question

${\int }_2^3\frac{\sqrt{x}}{\sqrt{(5-x)}+\sqrt{x}}dx+$

• A. $\frac{1}{2}$
• B. $\frac{1}{3}$
• C. $\frac{1}{5}$
• D. none

Answer 1

The correct option is A $\frac{1}{2}$

$I={\int }_2^3\frac{\sqrt{x}}{\sqrt{5-x}+\sqrt{x}}dx$ ...(1)

$I={\int }_2^3\frac{\sqrt{2+3-x}}{\sqrt{2+3-5+x}+\sqrt{2+3-x}}dx={\int }_2^3\frac{\sqrt{5-x}}{\sqrt{x}-\sqrt{5-x}}dx$ ...(2)

Adding (1) and (2), we get

$2I={\int }_2^{3}dx=1\Rightarrow I=\frac{1}{2}$