求不定积分(1+sin x)/(1+cos x)d(e^x)
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求不定积分(1+sin x)/(1+cos x)d(e^x)
![求不定积分(1+sin x)/(1+cos x)d(e^x)](/uploads/image/z/14921027-35-7.jpg?t=%E6%B1%82%E4%B8%8D%E5%AE%9A%E7%A7%AF%E5%88%86%281%2Bsin+x%29%2F%281%2Bcos+x%29d%28e%5Ex%29)
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
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
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