The value of $\int\limits_0^{\pi / 2} \frac{\sqrt{\sin x}}{\sqrt{\sin x}+\sqrt{\cos x}} d x$ is

$\frac{\pi}{4}$

Let $I=\int\limits_0^{\pi / 2} \frac{\sqrt{\sin x}}{\sqrt{\sin x}+\sqrt{\cos x}} d x$

Here the lower limit is zero hence, we can replace x by (a – x) i.e. by π/2 – x

∴ $I=\int\limits_0^{\pi / 2} \frac{\sqrt{\sin \left(\frac{\pi}{2}-x\right)}}{\sqrt{\sin \left(\frac{\pi}{2}-x\right)}+\sqrt{\cos \left(\frac{\pi}{2}-x\right)}} d x=\int\limits_0^{\pi / 2} \frac{\sqrt{\cos x}}{\sqrt{\cos x}+\sqrt{\sin x}} d x$

Adding $2 I=\int\limits_0^{\pi / 2} \frac{\sqrt{\sin x}+\sqrt{\cos x}}{\sqrt{\sin x}+\sqrt{\cos x}} d x=\frac{\pi}{2} \Rightarrow I=\frac{\pi}{4}$