# Primitive of Exponential Function/General Result/Proof 1

## 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

 $\ds \map {\dfrac \d {\d x} } {a^x}$ $=$ $\ds a^x \ln a$ 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)

$\blacksquare$