\( INT = \int_{0}^{\frac{\pi}{2}}\frac{1}{(1\!+x.\tan x)^2}\: dx= \int_{0}^{\frac{\pi}{2}}\frac{\cot^2x} {(\cot x + x)^2}\: dx\\ Stel \; \cot x + x = y\\ \scriptsize \quad (x\to 0\;\;k.o.m.\;y\to +\infty \quad x\to \frac{\pi}{2}\;k.o.m.\;y=\frac{\pi}{2})\\ \Rightarrow \left( - \; \frac {1}{\sin^2 x} + 1 \right) dx=dy\\ \Rightarrow \frac {\sin^2x\,-\: 1}{\sin^2x }dx=dy\\ \Rightarrow \cot^2x\;dx = -\:dy\\ INT= \int_{+\infty}^{\frac{\pi}{2}} \left( -\;\frac{1}{y^2}\right) \; dy=\left [ \frac 1y \right ]_{+\infty}^{\frac{\pi}{2}}=\frac{1}{\frac{\pi}{2}}=\frac{2}{\pi}\approx 0,63662 \)