Please find the indefinte integral S (e^(1/x) + 2) / x dx?

Question

S (e^(1/x) + 2) / x dx

I just need to see how this is worked out, thanks:)

Answer 1

Break the integral into 2 parts:
∫ e^(1/x) dx/x + 2∫ dx/x.
The second integral is just 2 ln (x).
Unfortunately, the first one is not elementary.
Let u = 1/x, x = 1/u dx = -1/u² du
Then it becomes
-∫ e^u du/u,
which is the nonelementary function -E_i(u) =-E_i(1/x_
the exponential integral.