Integral of Exponential function formula

Formula

$\displaystyle \int{a^{\displaystyle x} \,}dx \,=\, \dfrac{a^{\displaystyle x}}{\log_{e}{a}}+c$

Introduction

$a^{\displaystyle x}$ is an exponential function, where $a$ is a constant and $x$ is a variable. The integration of $a^{\displaystyle x}$ with respect to $x$ is expressed in mathematical form as follows.

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

The indefinite integral of $e^{\displaystyle x}$ with respect to $x$ is equal to the sum of the natural exponential function and constant of integration.

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

Other forms

The indefinite integration of natural exponential function formula can be written in terms of any variable.

$(1) \,\,\,$ $\displaystyle \int{a^{\displaystyle x} \,}dx \,=\, \dfrac{a^{\displaystyle x}}{\log_{e}{a}}+c$

$(2) \,\,\,$ $\displaystyle \int{a^{\displaystyle x} \,}dx \,=\, \dfrac{a^{\displaystyle x}}{\log_{e}{a}}+c$

$(3) \,\,\,$ $\displaystyle \int{a^{\displaystyle x} \,}dx \,=\, \dfrac{a^{\displaystyle x}}{\log_{e}{a}}+c$

Proof

Learn how to derive the indefinite integration rule for the natural exponential function in integral calculus.