Use the substitution u=1+e^x to find the Integral of e^(3x) / (1 + e^x)

e^x=u-1 so e^3x=(u-1)³ and du/dx = e^x so rearranging gives dx=e^-x du Substituting all that information in the integral we get Integral ( (u-1)³/ (u(u-1)) du ) which simplifies to Integral (u -2 +1/u).Integrating we get u²/2 -2u + ln u + C and substituting the original variable we get (1+e^x)²/2 -2(1+e^x) + ln (1+e^x) + C