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

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

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

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

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

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.

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

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

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

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

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

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

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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