The value of ∫limits_0^π / 2 n 1/(1+cot n x) d x, is

Let $I=\int\limits_0^{\pi / 2 n} \frac{1}{1+\cot n x} d x=\int\limits_0^{\pi / 2 n} \frac{\sin n x}{\cos n x+\sin n x} d x$

Putting $n x=t$, we get

$I=\frac{1}{n} \int\limits_0^{\pi / 2} \frac{\sin t}{\cos t+\sin t} d t=\frac{1}{n} \times \frac{\pi}{4}=\frac{\pi}{4 n}$