How to integrate (1/t) *( e^t )?

Question

I've tried solving it by parts but ended with a loop. And I can´t remember another method.

Thanks.

Answer 1

Unfortunately, there is no nice expression for it. The integral is often termed as the exponential integral and is not an elementary function, though you can represent it as a series as $$\int \dfrac{e^t}tdt = \int \sum_{k=0}^{\infty} \dfrac{t^{k-1}}{k!}dt = \ln(t) + \sum_{k=1}^{\infty} \dfrac{t^k}{k\cdot k!} + \text{constant}$$