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

Theorem
Let $n$ be a positive integer.

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

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

let:

and let:

Then: