Primitive of Exponential of a x
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Theorem
- $\ds \int e^{a x} \rd x = \frac {e^{a x} } a + C$
where $a$ is a non-zero constant.
Proof for Real Numbers
Let $x \in \R$ be a real variable.
\(\ds \int e^x \rd x\) | \(=\) | \(\ds e^x + C\) | Primitive of $e^x$ | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \int e^{a x} \rd x\) | \(=\) | \(\ds \frac 1 a \paren {e^{a x} } + C\) | Primitive of Function of Constant Multiple | ||||||||||
\(\ds \) | \(=\) | \(\ds \frac {e^{a x} } a + C\) | simplifying |
$\blacksquare$
Proof for Complex Numbers
Let $z \in \C$ be a complex variable.
\(\ds \map {D_x} {\frac {e^{a z} } a}\) | \(=\) | \(\ds \map {D_x} {\frac 1 a \sum_{n \mathop = 0}^\infty \frac {\paren {a z}^n} {n!} }\) | Definition of Complex Exponential Function | |||||||||||
\(\ds \) | \(=\) | \(\ds \map {D_x} {\frac 1 a \sum_{n \mathop = 0}^\infty \frac {a^n z^n} {n!} }\) | Exponent Combination Laws | |||||||||||
\(\ds \) | \(=\) | \(\ds \map {D_x} {\sum_{n \mathop = 0}^\infty \frac {a^{n - 1} z^n} {n!} }\) | Summation is Linear: Scaling of Summations | |||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{n \mathop = 1}^\infty n \frac {a^{n - 1} z^{n - 1} } {n!}\) | Derivative of Complex Power Series | |||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{n \mathop = 1}^\infty n \frac {\paren {a z}^{n - 1} } {n!}\) | Exponent Combination Laws | |||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{n \mathop = 1}^\infty \frac {\paren {a z}^{n - 1} } {\paren {n - 1}!}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{n \mathop = 0}^\infty \frac {\paren {a z}^n} {n!}\) | Translation of Index Variable of Summation | |||||||||||
\(\ds \) | \(=\) | \(\ds e^{a z}\) | Definition of Complex Exponential Function |
The result follows by the definition of the primitive.
$\blacksquare$
Sources
- 1944: R.P. Gillespie: Integration (2nd ed.) ... (previous) ... (next): Chapter $\text {II}$: Integration of Elementary Functions: $\S 7$. Standard Integrals: $3$.
- 1960: Margaret M. Gow: A Course in Pure Mathematics ... (previous) ... (next): Chapter $10$: Integration: $10.4$. Standard integrals: Standard Forms: $\text {(iii)}$
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $e^{a x}$: $14.509$
- 1968: George B. Thomas, Jr.: Calculus and Analytic Geometry (4th ed.) ... (previous) ... (next): Back endpapers: A Brief Table of Integrals: $102$.
- 1971: Wilfred Kaplan and Donald J. Lewis: Calculus and Linear Algebra ... (previous) ... (next): Appendix $\text I$: Table of Indefinite Integrals $8$.
- 1983: K.G. Binmore: Calculus ... (previous) ... (next): $9$ Sums and Integrals: $9.8$ 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
- 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