int (0 1) e^x dx/(1 + e^(2x)) = int (0 1) d(e^x)/(1 + e^(2x)) = | t = e^x | =
= int (1 e) dt/(1 + t^2) = (arctg t)_{1}^{e} = arctg e - arctg 1 = arctg e - pi/4.