Reduction Formula for Power of x by Exponential of a x
(Redirected from Primitive of Power of x by Exponential of a x/Lemma)
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Theorem
- $\ds \int x^n e^{a x} \rd x = \frac {x^n e^{a x} } a - \dfrac n a \int x^{n - 1} e^{a x} \rd x$
for $n \in \Z_{>0}$, $a \in \R_{\ne 0}$.
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:
\(\ds \frac {\d v} {\d x}\) | \(=\) | \(\ds e^{a x}\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds v\) | \(=\) | \(\ds \dfrac {e^{a x} } a\) | Primitive of $e^{a x}$ |
Then:
\(\ds \int x^n e^{a x} \rd x\) | \(=\) | \(\ds \int u \rd v\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \paren {x^n} \paren {\dfrac {e^{a x} } a} - \int \paren {\dfrac {e^{a x} } a} \paren {n x^{n - 1} } \rd x\) | 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 |
$\blacksquare$
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $e^{a x}$: $14.512$
- 1968: George B. Thomas, Jr.: Calculus and Analytic Geometry (4th ed.) ... (previous) ... (next): Back endpapers: A Brief Table of Integrals: $105$.
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Appendix: Table $3$: Reduction formulae
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Appendix: Table $3$: Reduction formulae