Primitive of Power of x by Exponential of a x/Lemma

Theorem
Let $n$ be a positive integer.

Then:
 * $\displaystyle \int x^n e^{a x} \ \mathrm d x = \frac {x^n e^{a x} } a - \frac n a \int x^{n - 1} e^{a x} \ \mathrm d x + C$

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: