拍悉伍袭或陆闷
.
∫arctan (e^x)/(e^x) dx
=-∫(arctan (e^x) de^(-x)
=-arctan(e^x) / e^x + ∫ dx/(1+e^(2x) )
let
e^x= tany
e^x dx = (secy)^2 dy
∫ dx/(1+e^(2x) )
= ∫ [1/乎肢(secy)^2] . [(secy)^2/仿念tany] dy
= ∫ (cosy/siny) dy
= ln|siny|+C'
= ln| e^x/√(1+e^2x) | + C'
∫(arctan e^x)/岁大世(e^x) dx
=- arctan(e^x) / e^x + ∫ dx/(1+e^(2x) )
=- arctan(e^x) / e^x + ln| e^x/√(1+e^2x) | + C
∫老空渗(arctan e^x)/(e^x)dx=∫(arctan e^x)d(e^x)=∫arctan udu,再用分部计分法，最侍脊后再还原亏瞎~