Question

求不定积分∫{[ln(e^x+1)]/e^x}dx

Answer 1

∫ln(e^x+1)dx/e^(x)

=-∫ln(e^x+1)de^(-x)

=-e^(-x)ln(e^x+1) +∫e^(-x)*(e^x)dx/(1+e^x)

=-e^(-x)ln(e^x+1)+∫dx/(1+e^x)

=-e^(-x)ln(e^x+1)+∫[1-e^x/(1+e^x)]dx

=-e^(-x)ln(e^x+1)+x-ln(e^x+1)+C