# Primitive of Exponential Function

Jump to navigation Jump to search

## 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 $e^{1 - x}$

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