求下列不定积分 ∫(arctan e^x)/(e^x)dx

题目

求下列不定积分 ∫(arctan e^x)/(e^x)dx

答案

∫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/(secy)^2] .[(secy)^2/tany] dy
= ∫ (cosy/siny) dy
= ln|siny|+C'
= ln| e^x/√(1+e^2x) | + C'
∫(arctan e^x)/(e^x) dx
=- arctan(e^x) / e^x + ∫ dx/(1+e^(2x) )
=- arctan(e^x) / e^x + ln| e^x/√(1+e^2x) | + C