定积分0到1,1/(1+e^x)dx

Question

定积分0到1,1/(1+e^x)dx

Answer 1

介于定积分很难书写,我就演示下算1/(1+e^x)的原函数
令t=e^x,x=lnt,dx=dt/t
∫1/(1+e^x)dx
=∫dt/(t^2+t)
=∫[1/t-1/(1+t)]dt
=ln|t|-ln|1+t|
=x-ln|1+e^x|+C
定积分0到1,1/(1+e^x)dx=(1-ln(1+e))-(0-ln2)=1+ln2-ln(1+e)