先化简被积函数=(1+2sin(x/2)cos(x/2))/2cos²(x/2)=1/2sec²(x/2)+tan(x/2)
故原式=1/2∫sec²(x/2)d(e^x)+∫tan(x/2)d(e^x)
=∫e^xd(tan(x/2))+∫tan(x/2)d(e^x)
=tan(x/2)e^x+C
求不定积分(1+sin x)/(1+cos x)d(e^x)
先化简被积函数=(1+2sin(x/2)cos(x/2))/2cos²(x/2)=1/2sec²(x/2)+tan(x/2)
故原式=1/2∫sec²(x/2)d(e^x)+∫tan(x/2)d(e^x)
=∫e^xd(tan(x/2))+∫tan(x/2)d(e^x)
=tan(x/2)e^x+C