Introduction to the integral of 2x

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

What is the integration of 2x?

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

Integral of 2x formula

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

$∫2xdx = x^2+ c$

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

How to calculate the integral of 2x?

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

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

Integral of 2x 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 (2x) by using u-substitution.

Proof of integration of 2x dx by using substitution method

To integrate 2x by using the substitution method, suppose that:

$I=\int 2xdx$

Suppose that,

$u=2x$

and

$du = 2 dx$

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

Now using this value in the above integral, 

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

We can write it as;

$I=\frac{1}{2}\int udu$

Integrating with respect to u,

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

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

Or, 

$I=\frac{u^2}{4}+c$

Now substituting the value of u,

$I=\frac{4x^2}{4}+c$

$I=x^2+c$

Hence we have verified the 2x integral by using the u-substitution method. 

2x integration 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 2x by using integration by parts.

Proof of integration of 2x by using integration by parts

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

$I=f(x)\int g(x)dx-\int[\frac{d}{dx}f'(x)\int g(x)dx]dx$

Suppose that, 

$f(x) = 2x$

$g(x) =1$ 

Using these values in the above formula, 

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

Integrating, 

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

$I=2x^2-\int 2xdx$

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

Simplifying,

$I=x^2+c$

Hence the integration of 2x is x^2 +c, where c is a constant known as an integration constant. To avoid above long calculations, use our by parts integration calculator.

Integral of 2x 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 integral of ln by using the indefinite integral.

Proof of integration of 2x dx by using definite integral

To integrate 2x by using a definite integral finder, we can use the interval from a to b. It means that we can evaluate the definite integration 2x for any value of a and b. Let’s compute it generally from a to b.

$∫^b_a 2x dx=x^2|^b_a$

Substituting the values of upper and lower bounds.

$∫^b_a 2x dx=b^2-a^2$

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 integral of x2?

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

$\int x^2dx=\frac{x^{2+1}}{2+1}=\frac{x^3}{3}+c$

What is integral formula?

The technique or method of finding an antiderivative of a function is known as integration. There are two types of integration formulas, definite and indefinite. The definite integral is used when a function is bounded on a specific interval. The indefinite integral is used when the function is not bounded. Mathematically, 

$\int f(x)dx$ (Indefinite integral)

$\int^b_a f(x)dx=F(b)-F(a)$ (Definite integral)