Question

同济,高数,208页 求不定积分积分 dx/(e^x+e^-x)

Answer 1

分子分母同时乘以 e^x
e^x/[(e^x)^2+1]dx
=1/[(e^x)^2+1] de^x
=arctan(e^x)+C
根据你给的应该是这样的

Answer 2

(e^x+e^-x)^-1
e^x=tant
e^x dx=sec²tdt
dx=sec²t/tant dt
dx=cost^(-2)*cost/sint dt
dx=dt/(sintcost)

换元后得
= ∫{ 1/(tant+cott)(sintcost) }dt
=∫{1/(sin²t+cos²t)} dt
=∫1dt
=t+C
t=arctan(e^x)
所以
结果=arctan(e^x)+C

Answer 3

都对 但是太麻烦 用双曲余切做简单
∫ dx/(e^x+e^-x)=∫ (sechx)/2dx
=[arctan(shx)]/2

Answer 4

楼上全部正解