Primitive of Exponential of a x

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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

$\blacksquare$


Proof for Complex Numbers

Let $z \in \C$ be a complex variable.

\(\displaystyle \map {D_x} {\frac {e^{a z} } a}\) \(=\) \(\displaystyle \map {D_x} {\frac 1 a \sum_{n \mathop = 0}^\infty \frac {\paren {a z}^n} {n!} }\) Definition of Complex Exponential Function
\(\displaystyle \) \(=\) \(\displaystyle \map {D_x} {\frac 1 a \sum_{n \mathop = 0}^\infty \frac {a^n z^n} {n!} }\) Exponent Combination Laws
\(\displaystyle \) \(=\) \(\displaystyle \map {D_x} {\sum_{n \mathop = 0}^\infty \frac {a^{n - 1} z^n} {n!} }\) Summation is Linear: Scaling of Summations
\(\displaystyle \) \(=\) \(\displaystyle \sum_{n \mathop = 1}^\infty n \frac {a^{n - 1} z^{n - 1} } {n!}\) Derivative of Complex Power Series
\(\displaystyle \) \(=\) \(\displaystyle \sum_{n \mathop = 1}^\infty n \frac {\paren {a z}^{n - 1} } {n!}\) Exponent Combination Laws
\(\displaystyle \) \(=\) \(\displaystyle \sum_{n \mathop = 1}^\infty \frac {\paren {a z}^{n - 1} } {\paren {n - 1}!}\)
\(\displaystyle \) \(=\) \(\displaystyle \sum_{n \mathop = 0}^\infty \frac {\paren {a z}^n} {n!}\) Translation of Index Variable of Summation
\(\displaystyle \) \(=\) \(\displaystyle e^{a z}\) Definition of Complex Exponential Function

The result follows by the definition of the primitive.

$\blacksquare$


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