求下列函数的不定积分 1/sqrt(1+e^x)

首先设e^x=t 则x=lnt
dx=1/tdt
1/sqrt(1+e^x) dx=1/[t*sqrt(1+t)]dt
在设sqrt(1+t)=m t=m^2-1
dt=2mdm
1/[t*sqrt(1+t)]dt=2m/[(m^2-1)*m]dm=2/[(m+1)(m-1)]dm
=[1/(m-1)-1/(m+1)]dm
=ln(m-1)-ln(m+1)
=ln[sqrt(1+e^x)-1]-ln[sqrt(1+e^x)+1]
=ln{[sqrt(1+e^x)-1]/[sqrt(1+e^x)+1]}+c