1、原式=∫ e^xdx/[(e^x)^2+1]
=∫ d(e^x)/[1+(e^x)^2]
=arctan(e^x)+C.
2、设x=sect,dx=sect*tantdt,
tant=√（x^2-1),
1/x=cost,
t=arccos(1/|x|)
原式=∫ tant*sect*tantdt/sect
=∫ （tant)^2dt
=∫ （[(sect)^2-1]dt
=tant-t+C
=√（x^2-1)-arccos(1/|x|)+C.