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.

Latest Math Topics

Latest Math Problems

Email subscription

Math Doubts is a free math tutor for helping students to learn mathematics online from basics to advanced scientific level for teachers to improve their teaching skill and for researchers to share their research projects.
Know more

Learn how to solve easy to difficult mathematics problems of all topics in various methods with step by step process and also maths questions for practising.