# Integral of X^2

Integral of x^2 along with its formula and proof with examples. Also learn how to calculate integration of x square with step by step examples.

Alan Walker-

Published on 2023-04-20

## Introduction to the integral of x^2

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 which is knwon 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 algebraic function x^2. You will also understand how to compute x^2 integral by using different integration techniques.

## What is the integral of x^2?

The integral of x2 is an antiderivative of x^2 function which is equal to x^3/3. It is also known as the reverse derivative of the function x^2 which is an algebraic function. It can be calculated by using the power rule of integral. This rule is written as;

$$\int x^ndx=\frac{x^{n+1}}{n+1}+c$$

This formula says that the integral of any algebraic function with some exponent, can be calculated by adding 1 in its exponent and dividing by the new exponent i.e n+1. The integral calculator with steps also uses this formula to evaluate integrals.

### Integral of x^2 formula

The formula of integral of x squared contains integral sign, coefficient of integration and the function as x^2. It is denoted by ∫(x^2)dx. In mathematical form, the integral of x^2 is:

$$∫x^2dx = \frac{x^3}{3}+ c$$

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

## How to calculate the integral of (x)^2?

The integral of x^2 is its antiderivative that can be calculated by using different integration techniques. In this article, we will discuss how to calculate integral of x square by using:

1. U-substitution method
2. Integration by parts
3. Definite integral

## Integral of x squared by using u-substitution method

The u substitution is a method of integration in calculus. It is used to calculate the integral of a function which is complex to be calculated by usual integration. Let’s discuss calculating the integral of (x)^2 by using u-substitution.

### Proof of Integral of x2 by using substitution method

To proof the integral of x^2 by using u-substitution calculator, suppose that:

$$I=\int x^2dx$$

Suppose that,

$$u=x^2$$

and

$$du = 2x dx$$

Since,

$$u=x^2$$

$$x=\sqrt{u}$$

So,

$$dx=\frac{du}{2u}$$

Now using this value in the above integral,

$$I=\int \frac{u}{2\sqrt{u}}du$$

We can write it as;

$$I=\frac{1}{2}\int \sqrt{u}du$$

Integrating with respect to u,

$$I=\frac{1}{2}\times \frac{u^{1/2+1}}{1/2+1}$$

$$I=\frac{1}{2}\frac{u^{3/2}}{3/2}$$

Or,

$$I=\frac{u^{3/2}}{3}+c$$

Now substituting the value of u,

$$I=\frac{(x^2)^{3/2}}{3}+c$$

$$I=\frac{x^3}{3}+c$$

Hence we have verified the x square integral by using the u-substitution method. We can also calculate the integral of sqrtx by using this method.

## Integral of x^2 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 integral of x^2 by using integration by parts.

### Proof of integral of (x)^2 by using integration by parts

To integrate the function x^2 using integration by parts, we can use the following formula:

$$I=f(x)\int g(x)dx-\int [f'(x)\int g(x)dx]dx$$

Suppose that,

$$f(x) = x^2$$

$$g(x) =1$$

Using these values in the above formula,

$$I=x^2\int 1dx-\int [\frac{d}{dx}(x^2)\int 1dx]dx$$

Integrating by using by parts integration calculator,

$$I=x^2.x-\int [2x.x]dx$$

$$I=x^3-2\int x^2dx$$

$$I=x^3-\frac{2x^3}{3}+c$$

Simplifying,

$$I=\frac{x^3}{3}+c$$

Hence the integral of x square is x^3/3 +c, where c is a constant known as an integration constant.

## Integral of x squared 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 formula can be written as:

$$∫^b_a f(x) dx = F(b) – F(a)$$

Let’s understand the verification of the integral of ln by using the indefinite integral.

### Proof of integral of log x by using definite integral

To compute the integral of ln by using a definite integral, we can use the interval from a to b. It means that we can evaluate the definite integral of x square for any value of a and b. Let’s compute it generally from a to b.

$$\int^b_a x^2dx=\frac{x^3}{3}|^b_a$$

Substituting the values of upper and lower bounds.

$$\int^b_a x^2dx=\frac{b^3}{3}-\frac{x^a}{3}$$

For any value of a and b, we can evaluate the definite integral by using the above formula. You can also use our definite integral calculator to make this calculation simple in just one click.

## Frequently Asked Questions

### What is the integration of x 4?

The antiderivative or the integral of x^4 is the same as the integral of x^2. It can be calculated as;

$$\int x^4 dx=\frac{x^{4+1}}{4+1}=\frac{x^5}{5}+c$$

### How do you find the integral?

To find the integral, we use the fundamental theorem of calculus. This theorem states that if a function f is continuous on an interval [a, b], then,

$$∫^b_a f(x) dx = F(b) – F(a)$$

Which is known as definite integral. Another formula to calculate the integral is the indefinite integral, that is;

$$∫f(x)=F(x) +c$$

### What is the integral of 1/(1+x^2)?

The integral 1/1+x^2 is equal to tan^-1(x). Mathematically, it is expressed as;

$$\int \frac{1}{1+x^2}dx=\tan^{-1}x+c$$

Where c is the constant of integration. The integral of 1/1+x^2 can also be calculated by using trigonometric substitution.