Question

∫(arctan e^x)/(e^x) dx ,e^x 是e的x次幂.

Answer 1

令e^x=t

则原式=∫arctant/t*1/tdt

=∫arctant/t^2dt

=-∫arctantd(1/t)

=-arctant/t+∫1/t*1/(t^2+1)dt

=-arctant/t+∫(1/t-t/(t^2+1))dt

=-arctant/t+∫dt/t-1/2∫d(t^2+1)/(t^2+1)

=-arctant/t+ln|t|-1/2ln|t^2+1|+C

=-arctan(e^x)/e^x+x-1/2ln(e^(2x)+1)+C