Proof of Integral of Natural Exponential function

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.

Derivative of Natural exponential function

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

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

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.

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

Integration of Natural exponential function

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.