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

Theorem

Then:

$\ds \int x^n e^{a x} \rd x = \frac {x^n e^{a x} } a - \frac n a \int x^{n - 1} e^{a x} \rd x + C$

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:

$\ds u$

$=$

$\ds x^n$

$\ds \leadsto \ \ $

$\ds \frac {\d u} {\d x}$

$=$

$\ds n x^{n - 1}$

and let:

$\ds \frac {\d v} {\d x}$

$=$

$\ds e^{a x}$

$\ds \leadsto \ \ $

$\ds v$

$=$

$\ds \frac {e^{a x} } a$

Then:

$\ds \int x^n e^{a x} \rd x$

$=$

$\ds x^n \frac {e^{a x} } a - \int \frac {e^{a x} } a \paren {n x^{n - 1} } \rd x + C$

$\ds $

$=$

$\ds \frac {x^n e^{a x} } a - \frac n a \int x^{n - 1} e^{a x} \rd x + C$

simplifying

$\blacksquare$