# Primitive of Exponential Function/General Result

## 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 1

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

$\blacksquare$

## Proof 2

Let $u = x \ln a$.

 $\ds \int a^x \rd x$ $=$ $\ds \int \map \exp {x \ln a} \rd x$ Definition of Power to Real Number $\ds$ $=$ $\ds \frac 1 {\ln a} \int \map \exp u \rd u$ Primitive of Function of Constant Multiple $\ds$ $=$ $\ds \frac {\map \exp u} {\ln a} + C$ Primitive of Exponential Function $\ds$ $=$ $\ds \frac {\map \exp {x \ln a} } {\ln a} + C$ Definition of $u$ $\ds$ $=$ $\ds \frac {a^x} {\ln a} + C$ Definition of Power to Real Number

$\blacksquare$