∫_π / 5^3 π / 10 (sin x)/(sin x+cos x) d x is equal to

Let $I=\int\limits_{\pi / 5}^{3 \pi / 10} \frac{\sin x}{\sin x+\cos x} d x$          ....(i)

Then, $I=\int\limits_{\pi / 5}^{3 \pi / 10} \frac{\sin \left(\frac{\pi}{2}-x\right)}{\sin \left(\frac{\pi}{2}-x\right)+\cos \left(\frac{\pi}{2}-x\right)} d x$

$\Rightarrow I=\int\limits_{\pi / 5}^{3 \pi / 10} \frac{\cos x}{\cos x+\sin x} d x$             ....(ii)

Adding (i) and (ii), we get

$2 I=\int\limits_{\pi / 5}^{3 \pi / 10} 1 d x=\frac{3 \pi}{10}-\frac{\pi}{5}=\frac{\pi}{10} \Rightarrow I=\frac{\pi}{20}$