Primitive of Power
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Theorem
Let $n \in \R: n \ne -1$.
Then:
- $\ds \int x^n \rd x = \frac {x^{n + 1} } {n + 1} + C$
where $C$ is an arbitrary constant.
That is:
- $\dfrac {x^{n + 1} } {n + 1}$ is a primitive of $x^n$.
Proof
\(\ds \map {\frac \d {\d x} } {\dfrac {x^{n + 1} } {n + 1} }\) | \(=\) | \(\ds \paren {n + 1} \paren {\dfrac {x^{\paren {n + 1} - 1} } {n + 1} }\) | Power Rule for Derivatives | |||||||||||
\(\ds \) | \(=\) | \(\ds x^n\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \int x^n \rd x\) | \(=\) | \(\ds \frac {x^{n + 1} } {n + 1} + C\) | Definition of Primitive (Calculus) |
When $n = -1$ we have $n + 1 = 0$, and $\dfrac {x^{n + 1} } {n + 1} = \dfrac {x^0} 0$ is undefined.
$\blacksquare$
Also known as
Some sources refer to this as the reverse power rule, as it is the "reverse" of the Power Rule for Derivatives.
It is even suggested that it could be called the anti-power rule, but this appears to be unlikely to catch on.
Also see
- Primitive of Reciprocal for the case where $n = -1$
Sources
- 1937: Eric Temple Bell: Men of Mathematics ... (previous) ... (next): Chapter $\text{VI}$: On the Seashore
- 1944: R.P. Gillespie: Integration (2nd ed.) ... (previous) ... (next): Chapter $\text {II}$: Integration of Elementary Functions: $\S 7$. Standard Integrals: $1$.
- 1960: Margaret M. Gow: A Course in Pure Mathematics ... (previous) ... (next): Chapter $10$: Integration: $10.4$. Standard integrals: Standard Forms: $\text {(i) (a)}$
- 1967: Michael Spivak: Calculus ... (previous) ... (next): Part $\text {III}$: Derivatives and Integrals: Chapter $18$: Integration in Elementary Terms
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: General Rules of Integration: $14.7$
- 1971: Wilfred Kaplan and Donald J. Lewis: Calculus and Linear Algebra ... (previous) ... (next): Appendix $\text I$: Table of Indefinite Integrals $5$.
- 1972: Frank Ayres, Jr. and J.C. Ault: Theory and Problems of Differential and Integral Calculus (SI ed.) ... (previous) ... (next): Chapter $25$: Fundamental Integration Formulas: $4$.
- 1974: Murray R. Spiegel: Theory and Problems of Advanced Calculus (SI ed.) ... (next): Chapter $5$. Integrals: Integrals of Special Functions: $1$
- 1983: K.G. Binmore: Calculus ... (next): $9$ Sums and Integrals: $9.8$ Standard Integrals
- 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): Appendix $2$: Table of derivatives and integrals of common functions
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Appendix: Table $2$: Integrals
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Appendix: Table $2$: Integrals
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): Appendix $7$: Integrals
- For a video presentation of the contents of this page, visit the Khan Academy.