设函数f(x)满足f(lnx) =ln(1+x)/x,求∫f(x)dx

设函数f(x)满足f(lnx) =ln(1+x)/x,求∫f(x)dx
设函数f(x)满足f(lnx) =ln(1+x)/x,求∫f(x)dx

设函数f(x)满足f(lnx) =ln(1+x)/x,求∫f(x)dx
令t = e^x,x = lnt,dx = (1/t)dt
∫ f(x) dx
= ∫ f(lnt) • (1/t)dt
= ∫ ln(1 + t)/t • (1/t)dt
= ∫ ln(1 + t) d(-1/t)
= (-1/t)ln(1 + t) + ∫ (1/t) • 1/(1 + t) dt
= (-1/t)ln(1 + t) + ∫ (1 + t - t)/[t(1 + t)] dt
= (-1/t)ln(1 + t) + ∫ [1/t - 1/(1 + t)] dt
= (-1/t)ln(1 + t) + ln|t| - ln|1 + t| + C
= (-1/e^x)ln(1 + e^x) + x - ln(1 + e^x) + C
= x - [1 + e^(-x)]•ln(1 + e^x) + C