A calculus problem by Srinivasa Gopal

$\begin{aligned} I & = \int_0^\frac \pi 2 \frac {\sqrt{\sin x}}{\sqrt{\sin x}+\sqrt{\cos x}} dx & \small \blue{\text{By reflection }\int_a^b f(x) \ dx = \int_a^b f(a+b-x)\ dx} \\ & = \frac 12 \int_0^\frac \pi 2 \left(\frac {\sqrt{\sin x}}{\sqrt{\sin x}+\sqrt{\cos x}} + \frac {\sqrt{\cos x}}{\sqrt{\cos x}+\sqrt{\sin x}} \right) dx \\ & = \frac 12 \int_0^\frac \pi 2 \frac {\sqrt{\sin x}+\sqrt{\cos x}}{\sqrt{\sin x}+\sqrt{\cos x}} dx \\ & = \frac 12 \int_0^\frac \pi 2 dx = \boxed {\frac \pi 4} \end{aligned}$

Reference: Reflections