# Integral of E^AX

Integral of e^ax along with its formula and proof with examples. Also learn how to calculate integration of e^ax with step by step examples.

Alan Walker-

Published on 2023-04-21

## Introduction to the integral of e^ax

In calculus, the integral is a fundamental concept that assigns numbers to functions to define displacement, area, volume, and all those functions that contain a combination of tiny elements. It is categorized into two parts, definite integral and indefinite integral. The process of integration calculates the integrals. This process is defined as finding an antiderivative of a function.

Integrals can handle almost all functions, such as trigonometric, algebraic, exponential, logarithmic, etc. This article will teach you what is integral to an exponential function e^ax. You will also understand how to compute the e^ax integral by using different integration techniques.

## What is the integral of e^ax?

The integral of e^(ax) is an antiderivative of the e^ax function which is equal to e^ax/a. The function e^ax is an exponential function and the integral of e is also known as its reverse derivative with respect to the variable involved. In simple words, we can say that finding the integral of e^ax is a process of reversing the derivative of e^ax.

### Integral of e^(ax) formula

The formula of the integral of e^ax contains the integral sign, coefficient of integration, and the function as sine. It is denoted by ∫(e^ax)dx. In mathematical form, the integral of e^ax is:

$∫e^{ax}dx = \frac{e^{ax}}{a} +c$

Where c is any constant involved, dx is the coefficient of integration and ∫ is the symbol of the integral. Use our integration calculator to evaluate the integration of e^(ax).

## How to calculate the integral of e^ax?

The integral of e^ax is its antiderivative that can be calculated by using different integration techniques. In this article, we will discuss how to calculate the integral of e^ax by using:

1. Integration by parts
2. Definite integral

## Integral of e by using derivatives

The derivative of a function calculates the rate of change, and integration is the process of finding the antiderivative of a function. Therefore, we can use the derivative to calculate the integral of a function. Let’s discuss calculating the integration of e^ax by using derivatives.

### Proof of integral of e^(ax) by using derivatives

Since we know that the integration is the reverse of the derivative. Therefore, we can calculate the integral of e^ax by using its derivative. For this, we have to look for some derivatives formulas or a formula that gives e^x as the derivative of any function.

In derivative, we know that,

$\frac{d}{dx}(e^{ax}) =ae^{ax}$

It means that the derivative of e^ax gives us ae^ax. Therefore, applying the integral on both sides,

$\int\frac{d}{dx}(e^{ax})dx=a\int e^{ax}dx$

The integral sign at the left side will cancel out with the derivative of e^ax and the resulting expression will be,

$a\int e^{ax}dx=e^{ax}+c$

Or,

$\int e^{ax}dx=\frac{e^{ax}}{a}+c$

Where c is a constant, known as the integration constant.

## Integration of e^ax by using integration by parts

The derivative of a function calculates the rate of change, and integration is the process of finding the antiderivative of a function. The integration by parts is a method of solving the integral of two functions combined together. Let’s discuss calculating the e^(ax) integral by using integration by parts.

### Proof of integral of e to the ax by using integration by parts

Since we know that the function e^x can be written as the product of two functions. Therefore, we can integrate it by using integration by parts. For this, suppose that:

$u = e^{ax}$

Taking the derivatives, we get:

$\frac{du}{dx} = e^{ax}$

Plugging these values into the integration by parts formula, we get:

$∫e^{ax} dx =1∫e^{ax} dx - ∫[\frac{d}{dx}(1)∫e^{ax} dx]dx$

Simplifying,

$∫e^{ax} dx =\frac{e^{ax}}{a} - 0$

$∫e^{ax} dx =\frac{e^{ax}}{a}$

where C is the constant of integration.

## Integral of e^ax by using definite integral

The definite integral is a type of integration that calculates the area of a curve by using infinitesimal area elements between two points. The definite integral can be written as:

$∫^b_a f(x) dx = F(b) – F(a)$

Let’s understand the verification of the integration of e^ax by using the definite integral.

### Proof of integral of e^ax by using definite integral

To compute the integral of e^ax by using a definite integral, we can use the interval from 0 to 1. Let’s compute the integral of e^(ax) from 0 to 1. For this we can write the integral as:

$∫^1_0 e^{ax} dx = \frac{e^{ax}}{a}|^1_0$

Now, substitute the limit in the given function.

$∫^1_0 e^{ax} dx = \frac{e^a}{a} – \frac{e^0}{a}=\frac{e^a}{a}-\frac{1}{a}$

For a = 1,

$∫^1_0 e^{ax} dx = e^1-1$

Since e^1 is equal to 2.718, therefore,

$∫^1_0 e^{ax} dx = 2.718 - 1=1.718$

Which is the same as the integral of e^x. Also, calculate the integral of e ax by using an indefinite integral finder.

## FAQ’s

### What is integration of E to the power ax?

The integration of e to the ax is the same as the integration of e^x. It is equal to the fraction of e^ax and the constant in its power i.e a. Mathematically, it can be expressed as;

$\int e^{ax}dx=\frac{e^{ax}}{a}+c$

### What is the value of e in the equation?

The function e is known as Euler’s number. Its value is a constant which is equal to 2.1782. It has many applications in calculus, physics, chemistry and many other fields of science.