Primitive of Exponential of a x over Power of x

Theorem

 * $\displaystyle \int \frac {e^{a x} \ \mathrm d x} {x^n} = \frac {-e^{a x} } {\left({n - 1}\right) x^{n - 1} } + \frac a {n - 1} \int \frac {e^{a x} \ \mathrm d x} {x^{n - 1} } + C$

where $n \ne 1$.

Proof
With a view to expressing the primitive in the form:
 * $\displaystyle \int u \frac {\mathrm d v}{\mathrm d x} \ \mathrm d x = u v - \int v \frac {\mathrm d u}{\mathrm d x} \ \mathrm d x$

let:

and let:

Then:

Also see

 * Primitive of $\dfrac {e^{a x} } x$ for the case where $n = 1$