Primitive of Exponential Function

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Theorem

$\displaystyle \int e^x \rd x = e^x + C$

where $C$ is an arbitrary constant.


General Result

Let $a \in \R_{>0}$ be a constant.

Then:

$\displaystyle \int a^x \ \mathrm d x = \frac {a^x} {\ln a} + C$

where $C$ is an arbitrary constant.


Proof for Real Numbers

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

\(\displaystyle \map {D_x} {e^x}\) \(=\) \(\displaystyle e^x\) Derivative of Exponential Function

The result follows by the definition of the primitive.

$\blacksquare$


Proof for Complex Numbers

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

\(\displaystyle D_z \left({e^z}\right)\) \(=\) \(\displaystyle e^z\) Definition of Complex Exponential Function

The result follows by the definition of the primitive.

$\blacksquare$


Also see


Sources