Primitive of Exponential Function/General Result/Proof 1
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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: $13$