设f(ln(x))=ln(X+1)/x,求f(x)的不定积分

Question

设f(ln(x))=ln(X+1)/x,求f(x)的不定积分

Answer 1

ƒ(lnx) = ln(x + 1)/x
令t = lnx,x = e^t
ƒ(t) = ln(1 + e^t)/e^t
∫ ƒ(x) dx
= ∫ ln(1 + e^x)/e^x dx
= - ∫ ln(1 + e^x) d[e^(- x)]
= - ln(1 + e^x)/e^x + ∫ 1/[e^x(1 + e^x)] d(e^x)
= - ln(1 + e^x)/e^x + ∫ [(1 + e^x) - e^x]/[e^x(1 + e^x)] d(e^x)
= - ln(1 + e^x)/e^x + ∫ [1/e^x - 1/(1 + e^x)] d(e^x)
= - ln(1 + e^x)/e^x + ln(e^x) - ln(1 + e^x) + C
= - ln(1 + e^x)/e^x + ln[e^x/(1 + e^x)] + C
= - ln(1 + e^x)/e^x - ln[e^(- x) + 1] + C