Integración por cambio de variable

∫ [e^x/ (e^2x + 2e^x + 1)] dx

u = e^x

du = e^x dx

dx = du/ e^x

ln u = ln e^x

ln u = x

∫[u/ (u² + 2u + 1)](du/ u) = ∫ du/ (u² + 2u + 1) = ∫ du/ (u + 1)² = [- 1/ (u + 1)] + c

∫ [e^x/ (e^2x + 2e^x + 1)] dx = [- 1/ (e^x + 1)] + c