Evaluate ∫₀^π/2 √(sin x) √(sin x) + √(cos x) , dx — JEE Mathematics

Let $I = \int_{0}^{\pi/2} \frac{\sqrt{\sin x}}{\sqrt{\sin x} + \sqrt{\cos x}} \, dx \quad \dots (1)$
Apply the identity $\int_{0}^{a} f(x) \, dx = \int_{0}^{a} f(a-x) \, dx$:

$$I = \int_{0}^{\pi/2} \frac{\sqrt{\sin(\pi/2 - x)}}{\sqrt{\sin(\pi/2 - x)} + \sqrt{\cos(\pi/2 - x)}} \, dx = \int_{0}^{\pi/2} \frac{\sqrt{\cos x}}{\sqrt{\cos x} + \sqrt{\sin x}} \, dx \quad \dots (2)$$

Add equations (1) and (2):

$$2I = \int_{0}^{\pi/2} \frac{\sqrt{\sin x} + \sqrt{\cos x}}{\sqrt{\sin x} + \sqrt{\cos x}} \, dx = \int_{0}^{\pi/2} 1 \, dx$$

$$2I = [x]_{0}^{\pi/2} = \frac{\pi}{2} \implies I = \frac{\pi}{4}$$

Answer: $\frac{\pi}{4}$