∫[(1+sinx)/(1+cosx)]*(e^x)dx
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![∫[(1+sinx)/(1+cosx)]*(e^x)dx](/uploads/image/z/6863931-27-1.jpg?t=%E2%88%AB%5B%281%2Bsinx%29%2F%281%2Bcosx%29%5D%2A%28e%5Ex%29dx)
∫[(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