Step-by-step Solution:

We are given the integral: \[ I = \int_{0}^{\pi/2} \frac{\sqrt{\sin x}}{\sqrt{\sin x} + \sqrt{\cos x}} \, dx \] As we know that: \[ \int_{0}^{a} f(x) \, dx = \int_{0}^{a} f(a - x) \, dx \] Step 1:Using above we get: \[ I = \int_{0}^{\pi/2} \frac{\sqrt{\sin \left(\frac{\pi}{2} - x \right)}}{\sqrt{\sin \left(\frac{\pi}{2} - x \right)} + \sqrt{\cos \left(\frac{\pi}{2} - x \right)}} \, dx \] Simplifying: \[ I = \int_{0}^{\pi/2} \frac{\sqrt{\cos x}}{\sqrt{\cos x} + \sqrt{\sin x}} \, dx \] Step 2: Add the two equations Adding the original integral and the modified integral: \[ 2I = \int_{0}^{\pi/2} \left( \frac{\sqrt{\sin x}}{\sqrt{\sin x} + \sqrt{\cos x}} + \frac{\sqrt{\cos x}}{\sqrt{\cos x} + \sqrt{\sin x}} \right) \, dx \] Simplifying the integrand: \[ 2I = \int_{0}^{\pi/2} 1 \, dx \] Step 3: Evaluate the integral Evaluating the integral: \[ 2I = \left[ x \right]_{0}^{\pi/2} = \frac{\pi}{2} \] Thus: \[ I = \frac{\pi}{4} \]