Reduction Formula for Primitive of Product of Power with Exponential
Jump to navigation
Jump to search
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
- 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): Appendix $8$: Integrals: Reduction Formulae