## Introduction to the integral of e^3x

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^3x. You will also understand how to compute the e^3x integral by using different integration techniques.

## What is the integral of e^3x?

The integral of e^(3x) is an antiderivative of the e^3x function which is equal to e^3x/3. The function e^3x 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 antiderivative of e 3x is a process of reversing the derivative of e^3x.

### E^3x integration formula

The formula of the integral of e 3x contains the integral sign, coefficient of integration, and the function as sine. It is denoted by ∫(e^3x)dx. In mathematical form, the integral of e^3x is:

$∫(e^3x)dx = \frac{e^{3x}}{3} +c$

Where c is any constant involved, dx is the coefficient of integration and ∫ is the symbol of the integral.

## How to evaluate integral of e^(3x) with respect to x?

The e^3x integration 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^3x by using:

- Integration by parts
- 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 compute the integral of a function. Let’s discuss calculating the integration of e^3x by using derivatives.

**Proof of integral of e^(3x) by using derivatives**

Since we know that the integration is the reverse of the derivative. Therefore, we can integrate e^3x 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,

$\frac{d}{dx}(e^{3x}) =3e^{3x}$

It means that the derivative of e^3x gives us ae^3x. Therefore, applying the integral on both sides,

$\frac{d}{dx}(e^{3x})dx=3\int e^{3x}dx

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

$3\int e^{3x}=e^{3x}$

Or,

$\int e^{3x}dx=\frac{e^{3x}}{3}+c$

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

**Integration of e^3x 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^(3x) integral by using integration by parts calculator.

**Proof of integral of e to the 3x 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:

$u = e^{3x}$

Taking the derivatives, we get:

$\frac{du}{dx} = e^{3x}$

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

$∫e^{3x} dx =1∫e^{3x} dx - ∫[\frac{d}{dx}(1)∫e^{3x} dx]dx$

Simplifying,

$∫e^{3x} dx =\frac{e^{3x}}{3} - 0$

$∫e^{3x} dx =\frac{e^{3x}}{3} + c$

where C is the constant of integration.

## Integral of e^3x by using definite integral

The definite integral is a type of integral 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^3x by using the definite integral.

### Proof of antiderivative of e 3x by using definite integral

To evaluate integral of e^(3x) with respect to x by using a definite integral, we can use the interval from 0 to 1. Let’s compute the antiderivative of e 3x from 0 to 1. For this, we can write the integral as:

$∫^1_0 e^{3x} dx = \frac{e^{3x}}{3}|^1_0$

Now, substitute the limit in the given function.

$∫^1_0 e^{3x} dx = \frac{e^{3}}{3} – \frac{e^0}{3}=\frac{e^3}{3}-\frac{1}{3}$

Since e^3=20

$∫^1_0 e^{3x} dx = \frac{20}{3}-\frac{1}{3}$

$∫^1_0 e^{3x} dx = 6.361$

You can also use our definite integration calculator to evaluate the definite integral of e3x.

## FAQ’s

### What is integration of 4x?

The integration of e to the 4x is the same as the integration of e^3x. It is equal to the fraction of e^4x and the constant in its power i.e 4. Mathematically, it can be expressed as;

$\int e^{4x}dx=\frac{e^{4x}}{4}+c$

### How do I evaluate the definite integral of e^3x from a to b?

You can evaluate the definite integral of e 3x by using the following formula,

$∫^b_a e^{3x} dx = F(b) – F(a)$

For any value of a and b, you can evaluate the above equation.