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.
A best free mathematics education website for students, teachers and researchers.
Learn each topic of the mathematics easily with understandable proofs and visual animation graphics.
Learn how to solve the math problems in different methods with understandable steps and worksheets on every concept for your practice.
Copyright © 2012 - 2022 Math Doubts, All Rights Reserved