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.

Integration of x2 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 integration x2 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.

Integration of x2 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 x 2 by using definite integral

To compute the integral of x2 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.