Primitive of Exponential Function/General Result/Proof 2
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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
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
- 1944: R.P. Gillespie: Integration (2nd ed.) ... (previous) ... (next): Chapter $\text {II}$: Integration of Elementary Functions: $\S 7$. Standard Integrals: Notes