求积分∫ 1/(1+e^2x) dx
问题描述：
求积分∫ 1/(1+e^2x) dx
答
=lim(x→1）(x-1)(x^2+x+1)/[(x-1)(x+1)] =lim(x→1）(x^2+x+1)/(x+1) =3/2 ∫[0,1]x*e^(2x)dx （用分步积分法） =1
答
设t=e^(2x),x=(lnt)/2,dx=1/(2t) dt
∫dx/[1+e^(2x)]
= (1/2)∫dt/[t(1+t)]
= (1/2)∫[(1+t)-t]/[t(1+t)] dt
= (1/2)∫[1/t - 1/(1+t)] dt
= (1/2)[ln|t| - ln|1+t|] + C
= (1/2)[ln|e^(2x)| - ln|1+e^(2x)] + C
= x - (1/2)ln|1+e^(2x)| + C