# 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:

 $\displaystyle u$ $=$ $\displaystyle x^n$ $\displaystyle \leadsto \ \$ $\displaystyle \frac {\d u} {\d x}$ $=$ $\displaystyle n x^{n - 1}$ Derivative of Power

and let:

 $\displaystyle \frac {\d v} {\d x}$ $=$ $\displaystyle e^{a x}$ $\displaystyle \leadsto \ \$ $\displaystyle v$ $=$ $\displaystyle \frac {e^{a x} } a$ Primitive of Exponential of a x

Then:

 $\displaystyle \int x^n e^{a x} \rd x$ $=$ $\displaystyle x^n \frac {e^{a x} } a - \int \frac {e^{a x} } a \paren {n x^{n - 1} } \rd x + C$ Integration by Parts $\displaystyle$ $=$ $\displaystyle \frac {x^n e^{a x} } a - \frac n a \int x^{n - 1} e^{a x} \rd x + C$ simplifying

$\blacksquare$