what is the antiderivatie of (e^x)/x?

Answer 1

[0, x]∫e^t/t dt

That's really as simple as you can make it -- this integral cannot be expressed in terms of elementary functions.

Re: allseeinggirl -- If you try to apply integration by parts to this using the variables, u=ln x, du=1/x dx, v=e^x, dv=e^x dx, you obtain ∫e^x/x dx = e^x ln x - ∫e^x ln x dx, not e^x ln x - ∫e^x/x dx as you incorrectly stated. Also, (e^x ln x)/2 cannot be the antiderivative of e^x/x, because its derivative is (e^x ln x)/2 + e^x/(2x).

Answer 2

Integration by Parts

u=ln(x)
du=1/x dx
v=e^x
dv=e^x dx

=ln(x)e^x - INTEGRAL of (e^x/x)

INTEGRAL of e^x/x + INTEGRAL of e^x/x =ln(x)e^x

INTEGRAL of e^x/x = (ln(x)e^x)/2