Reduction Formula for Primitive of Product of Power with Exponential

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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:

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


and let:

\(\ds \frac {\d v} {\d x}\) \(=\) \(\ds e^x\)
\(\ds \leadsto \ \ \) \(\ds v\) \(=\) \(\ds e^x\) Derivative of Exponential Function


Then:

\(\ds I_n\) \(=\) \(\ds \int x^n e^x \rd x\) by definition
\(\ds \) \(=\) \(\ds x^n e^x - \int n x^{n - 1} e^x \rd x\) Integration by Parts
\(\ds \) \(=\) \(\ds x^n e^x - n I_{n - 1}\)

$\blacksquare$


Sources

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