Primitive of Power/Proof

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.

$\ds \map {\frac \d {\d x} } {\dfrac {x^{n + 1} } {n + 1} }$

$=$

$\ds \paren {n + 1} \paren {\dfrac {x^{\paren {n + 1} - 1} } {n + 1} }$

$\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.

$\blacksquare$

1944: R.P. Gillespie: Integration (2nd ed.) ... (previous) ... (next): Chapter $\text {II}$: Integration of Elementary Functions: $\S 7$. Standard Integrals: $1$.
1945: A. Geary, H.V. Lowry and H.A. Hayden: Advanced Mathematics for Technical Students, Part I ... (previous) ... (next): Chapter $\text {III}$: Integration: Integration
1953: L. Harwood Clarke: A Note Book in Pure Mathematics ... (next): $\text {II}$. Calculus: Integration
1967: Michael Spivak: Calculus ... (previous) ... (next): Part $\text {III}$: Derivatives and Integrals: Chapter $18$: Integration in Elementary Terms
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
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