Primitive of Exponential Function/General Result/Proof 1

Theorem

Let $a \in \R_{>0}$ be a constant such that $a \ne 1$.

Then:

$\ds \int a^x \rd x = \frac {a^x} {\ln a} + C$

where $C$ is an arbitrary constant.


Proof

\(\ds \map {\dfrac \d {\d x} } {a^x}\) \(=\) \(\ds a^x \ln a\) Derivative of General Exponential Function
\(\ds \leadsto \ \ \) \(\ds \map {\dfrac \d {\d x} } {\dfrac {a^x} {\ln a} }\) \(=\) \(\ds a^x\) Derivative of Constant Multiple
\(\ds \leadsto \ \ \) \(\ds \int a^x \rd x\) \(=\) \(\ds \dfrac {a^x} {\ln a}\) Definition of Primitive (Calculus)

$\blacksquare$


Sources

  • 1953: L. Harwood Clarke: A Note Book in Pure Mathematics ... (previous) ... (next): $\text {II}$. Calculus: Integration
  • 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: $6$.
  • 1974: Murray R. Spiegel: Theory and Problems of Advanced Calculus (SI ed.) ... (previous) ... (next): Chapter $5$. Integrals: Integrals of Special Functions: $13$
This article is issued from ProofWiki. The text is available under Creative Commons Attribution-ShareAlike License unless otherwise noted. Additional terms may apply for the media files.