∫[(1+sinx)/(1+cosx)]*(e^x)dx

∫[(1+sinx)/(1+cosx)]*(e^x)dx
∫[(1+sinx)/(1+cosx)]*(e^x)dx

∫[(1+sinx)/(1+cosx)]*(e^x)dx
∫[(1+sinx)/(1+cosx)]*(e^x)dx
=∫[ (1+2sin(x/2)cos(x/2)) / (2cos²(x/2)) ]*(e^x)dx
=∫[ (1/2)sec²(x/2)+tan(x/2) ]*(e^x)dx
=(1/2)∫ sec²(x/2)e^x dx+ ∫ (e^x)tan(x/2) dx
=∫ sec²(x/2)e^x d(x/2)+ ∫ (e^x)tan(x/2) dx
=∫ e^x d(tan(x/2))+ ∫ (e^x)tan(x/2) dx
前一项用分部积分
=(e^x)tan(x/2)-∫ (e^x)tan(x/2)dx+ ∫ (e^x)tan(x/2) dx
=(e^x)tan(x/2)+C