求不定积分∫{[ln(e^x+1)]/e^x}dx
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求不定积分∫{[ln(e^x+1)]/e^x}dx
![求不定积分∫{[ln(e^x+1)]/e^x}dx](/uploads/image/z/15977268-36-8.jpg?t=%E6%B1%82%E4%B8%8D%E5%AE%9A%E7%A7%AF%E5%88%86%E2%88%AB%7B%5Bln%28e%5Ex%2B1%29%5D%2Fe%5Ex%7Ddx)
∫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
=-∫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