# Primitive of Exponential of a x

## Theorem

$\displaystyle \int e^{a x} \rd x = \frac {e^{a x} } a + C$

## Proof for Real Numbers

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

 $\displaystyle \int e^x \rd x$ $=$ $\displaystyle e^x + C$ Primitive of $e^x$ $\displaystyle \leadsto \ \$ $\displaystyle \int e^{a x} \rd x$ $=$ $\displaystyle \frac 1 a \paren {e^{a x} } + C$ Primitive of Function of Constant Multiple $\displaystyle$ $=$ $\displaystyle \frac {e^{a x} } a + C$ simplifying

$\blacksquare$

## Proof for Complex Numbers

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

 $\displaystyle \map {D_x} {\frac {e^{a z} } a}$ $=$ $\displaystyle \map {D_x} {\frac 1 a \sum_{n \mathop = 0}^\infty \frac {\paren {a z}^n} {n!} }$ Definition of Complex Exponential Function $\displaystyle$ $=$ $\displaystyle \map {D_x} {\frac 1 a \sum_{n \mathop = 0}^\infty \frac {a^n z^n} {n!} }$ Exponent Combination Laws $\displaystyle$ $=$ $\displaystyle \map {D_x} {\sum_{n \mathop = 0}^\infty \frac {a^{n - 1} z^n} {n!} }$ Summation is Linear: Scaling of Summations $\displaystyle$ $=$ $\displaystyle \sum_{n \mathop = 1}^\infty n \frac {a^{n - 1} z^{n - 1} } {n!}$ Derivative of Complex Power Series $\displaystyle$ $=$ $\displaystyle \sum_{n \mathop = 1}^\infty n \frac {\paren {a z}^{n - 1} } {n!}$ Exponent Combination Laws $\displaystyle$ $=$ $\displaystyle \sum_{n \mathop = 1}^\infty \frac {\paren {a z}^{n - 1} } {\paren {n - 1}!}$ $\displaystyle$ $=$ $\displaystyle \sum_{n \mathop = 0}^\infty \frac {\paren {a z}^n} {n!}$ Translation of Index Variable of Summation $\displaystyle$ $=$ $\displaystyle e^{a z}$ Definition of Complex Exponential Function

The result follows by the definition of the primitive.

$\blacksquare$