Reduction Formula for Power of x by Exponential of a x

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

for $n \in \Z_{>0}$, $a \in \R_{\ne 0}$.


With a view to expressing the primitive in the form:

$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$


\(\ds u\) \(=\) \(\ds x^n\)
\(\ds \leadsto \ \ \) \(\ds \frac {\d u} {\d x}\) \(=\) \(\ds n x^{n - 1}\) Power Rule for Derivatives

and let:

\(\ds \frac {\d v} {\d x}\) \(=\) \(\ds e^{a x}\)
\(\ds \leadsto \ \ \) \(\ds v\) \(=\) \(\ds \dfrac {e^{a x} } a\) Primitive of $e^{a x}$


\(\ds \int x^n e^{a x} \rd x\) \(=\) \(\ds \int u \rd v\)
\(\ds \) \(=\) \(\ds \paren {x^n} \paren {\dfrac {e^{a x} } a} - \int \paren {\dfrac {e^{a x} } a} \paren {n x^{n - 1} } \rd x\) Integration by Parts
\(\ds \) \(=\) \(\ds \frac {x^n e^{a x} } a - \dfrac n a \int x^{n - 1} e^{a x} \rd x\) simplifying