Primitive of Exponential Function/General Result

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 1

$\ds \map {\dfrac \d {\d x} } {a^x}$

$=$

$\ds a^x \ln a$

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)

$\blacksquare$

Proof 2

Let $u = x \ln a$.

$\ds \int a^x \rd x$

$=$

$\ds \int \map \exp {x \ln a} \rd x$

Definition of Power to Real Number

$\ds $

$=$

$\ds \frac 1 {\ln a} \int \map \exp u \rd u$

Primitive of Function of Constant Multiple

$\ds $

$=$

$\ds \frac {\map \exp u} {\ln a} + C$

Primitive of Exponential Function

$\ds $

$=$

$\ds \frac {\map \exp {x \ln a} } {\ln a} + C$

Definition of $u$

$\ds $

$=$

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

Definition of Power to Real Number

$\blacksquare$

Sources

1953: L. Harwood Clarke: A Note Book in Pure Mathematics ... (previous) ... (next): $\text {II}$. Calculus: Integration
1960: Margaret M. Gow: A Course in Pure Mathematics ... (previous) ... (next): Chapter $10$: Integration: $10.4$. Standard integrals: Other Standard Results: $\text {(xii)}$
1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: General Rules of Integration: $14.10$
1968: George B. Thomas, Jr.: Calculus and Analytic Geometry (4th ed.) ... (previous) ... (next): Front endpapers: A Brief Table of Integrals: $2$.
1971: Wilfred Kaplan and Donald J. Lewis: Calculus and Linear Algebra ... (previous) ... (next): Appendix $\text I$: Table of Indefinite Integrals $9$.
1976: K. Weltner and W.J. Weber: Mathematics for Engineers and Scientists ... (previous) ... (next): $6$. Integral Calculus: Appendix: Table of Fundamental Standard Integrals
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
2009: Murray R. Spiegel, Seymour Lipschutz and John Liu: Mathematical Handbook of Formulas and Tables (3rd ed.) ... (previous) ... (next): $\S 16$: Indefinite Integrals: General Rules of Integration: $16.10.$
2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): Appendix $7$: Integrals
2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): Appendix $8$: Integrals