Primitive of Exponential of a x/Real

Theorem

$\ds \int e^{a x} \rd x = \frac {e^{a x} } a + C$

Proof for Real Numbers

Let $x \in \R$ be a real variable.

\(\ds \int e^x \rd x\)

\(=\)

\(\ds e^x + C\)

Primitive of $e^x$

\(\ds \leadsto \ \ \)

\(\ds \int e^{a x} \rd x\)

\(=\)

\(\ds \frac 1 a \paren {e^{a x} } + C\)

Primitive of Function of Constant Multiple

\(\ds \)

\(=\)

\(\ds \frac {e^{a x} } a + C\)

simplifying

$\blacksquare$