Reduction Formula for Primitive of Product of Power with Exponential

Theorem
Let $n \in \Z_{> 0}$ be a (strictly) positive integer.

Let:
 * $I_n := \ds \int x^n e^x \rd x$

Then:
 * $I_n = x^n e^x - n I_{n - 1}$

is a reduction formula for $\ds \int x^n e^x \rd x$.

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

let:

and let:

Then: