Primitive of Exponential Function/General Result

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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

$\blacksquare$


Sources

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