Primitive of Exponential of a x over x

Theorem

 * $\displaystyle \int \frac {e^{a x} \, \mathrm d x} x = \ln \left\vert{x}\right\vert + \sum_{k \mathop \ge 1} \frac {\left({a x}\right)^k} {k \times k!} + C$

Proof
The validity of $(1)$ follows from absolute convergence of the power series expansion.

Also see

 * Primitive of $\dfrac {e^x} x$ has no Solution in Elementary Functions