# Proof of Integral of Natural Exponential function

Assume, $x$ is a variable and the natural exponential function is written as $e^x$ in mathematics. The indefinite integration of natural exponential function with respect to $x$ is written in the following mathematical form in integral calculus.

$\displaystyle \int{e^x} \,}d$

Now, let us learn how to derive the proof for the integration rule of the natural exponential function.

### Derivative of Natural exponential function

Write the formula for the derivative of natural exponential function with respect to $x$ in mathematical form.

$\dfrac{d}{dx}{\, (e^x})} \,=\, e^x$

### Inclusion of an Arbitrary constant

Include a constant to natural exponential function but it does not change the differentiation of sum of natural exponential function and constant because the derivative of a constant is zero.

$\implies$ $\dfrac{d}{dx}{\, (e^x}+c)} \,=\, e^x$

### Integration of Natural exponential function

According to the integration, the collection of all primitives of $e^x$ function is called the integration of $e^x$ function with respect to $x$. It is expressed in mathematics as follows.

$\displaystyle \int{e^x} \,}d$

The antiderivative or primitive of $e^x$ function is sum of the natural exponential function and the constant of integration ($c$).

$\dfrac{d}{dx}{(e^x}+c)} = e^x$ $\,\Longleftrightarrow\,$ $\displaystyle \int{e^x} \,}dx = e^x}+$

$\therefore \,\,\,\,\,\,$ $\displaystyle \int{e^x} \,}dx \,=\, e^x}+$

Therefore, it has proved that the integration of natural exponential function with respect to a variable is equal to the sum of the natural exponential function and the constant of integration.

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