Value of $\int\limits_2^3\frac{\sqrt{x}}{\sqrt{x}+\sqrt{5-x}}dx$ is

$\frac{1}{2}$

The correct answer is Option (2) → $\frac{1}{2}$

$\int_{2}^{3} \frac{\sqrt{x}}{\sqrt{x} + \sqrt{5 - x}}\,dx$

Let $I = \int_{2}^{3} \frac{\sqrt{x}}{\sqrt{x} + \sqrt{5 - x}}\,dx$

Use the identity: $\int_{a}^{b} f(x)\,dx = \int_{a}^{b} f(a + b - x)\,dx$

$I = \int_{2}^{3} \frac{\sqrt{5 - x}}{\sqrt{5 - x} + \sqrt{x}}\,dx$

Add both expressions:

$2I = \int_{2}^{3} \left( \frac{\sqrt{x}}{\sqrt{x} + \sqrt{5 - x}} + \frac{\sqrt{5 - x}}{\sqrt{5 - x} + \sqrt{x}} \right) dx$

$2I = \int_{2}^{3} 1\,dx = 3 - 2 = 1$

$I = \frac{1}{2}$