Primitive of Exponential of a x over x

Theorem

 * $\ds \int \frac {e^{a x} \rd x} x = \ln \size x + \sum_{k \mathop \ge 1} \frac {\paren {a x}^k} {k \times k!} + C$

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