Primitive of Exponential Function

Theorem

$\ds \int e^x \rd x = e^x + C$

where $C$ is an arbitrary constant.


General Result

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 for Real Numbers

Let $x \in \R$ be a real variable.

\(\ds \map {\dfrac \d {\d x} } {e^x}\) \(=\) \(\ds e^x\) Derivative of Exponential Function

The result follows by the definition of the primitive.

$\blacksquare$


Proof for Complex Numbers

Let $z \in \C$ be a complex variable.

\(\ds \map {D_z} {e^z}\) \(=\) \(\ds e^z\) Definition of Complex Exponential Function

The result follows by the definition of the primitive.

$\blacksquare$


Examples

Primitive of $e^{1 - x}$

$\ds \int e^{1 - x} \rd x = -e^{1 - x} + C$


Also see


Sources

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  • 1953: L. Harwood Clarke: A Note Book in Pure Mathematics ... (previous) ... (next): $\text {II}$. Calculus: Integration
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  • 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: General Rules of Integration: $14.9$
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