[tex] \int\limits^1_0 { e^{x}/ (e^{2x}+1) } \, dx [/tex]
Notam e^x=t => e^2x=t^2
Daca e^x=t si logaritmam cu logaritm natural avem: x=ln t => dx=dt/t
Daca x=0 => t=e^0=1
Daca x=1 => t=e^1=e
Si avem:
[tex] \int\limits^e_1 {t/t( t^{2}+1 )} \, dt = \int\limits^e_1 {1/ t^{2}+ 1} \, dt=arctg~t |_{1} ^{e}=arctg~e-arctg~1[/tex]=arctg e-pi/4
