Primitive of Power/Proof

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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.


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.

$\blacksquare$


Sources

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