Primitive of Exponential of a x/Real

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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

$\blacksquare$