Integral of Sqrt x

Integral of sqrtx along with its formula and proof with examples. Also learn how to calculate integration of sqrt x with step by step examples.

Alan Walker-

Published on 2023-04-21

Introduction to the integral of square root x

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. 

Integration calculators 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. You will also understand how to compute √x integral by using different integration techniques.

What is the integral of sqrt x?

The integral of √x is an antiderivative of sqrt x function which is equal to 2/3x^(3/2). It is also known as the reverse derivative of the function √x which is an algebraic function.

This integral sqrt x can be calculated by using the power rule of the integral. This rule is written as;

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

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.

Integration of √x formula

The formula of integral of square root x squared contains the integral sign, coefficient of integration, and the function as √x. It is denoted by ∫√x dx. In mathematical form, the integral sqrt x is:

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

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

How to calculate the integral of √x?

The integral of sqrt(x) is its antiderivative, which can be calculated using different integration techniques. In this article, we will discuss how to calculate integral of a square root x 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 that is complex to be calculated by usual integration. Let’s discuss calculating the integral of square root x by using u-substitution calculator.

Proof of antiderivative of sqrt x by using substitution method

To prove the integral of √x by using the substitution method, suppose that:


Suppose that,




Since u=√x,



Now using this value in the above integral, 

$I=\int 2u^2du$

We can write it as;

$I=2\int u^2du$

Integrating with respect to u,





Now substituting the value of u,



Hence we have verified the integration of √x formula by using the u-substitution method.

Integral of sqrt x 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 sqrt(x) by using integration by parts.

Proof of integration of root x formula by using integration by parts

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

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

Suppose that, 

$f(x) = √x$

$g(x) =1$ 

Using these values in the above formula, 

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


$I=\sqrt{x}x-\int[\frac{1}{2\sqrt{x}}\times x]dx$

$I=x^{3/2}-2\int \frac{x}{\sqrt{x}}dx$

Simplifying by rationalizing x/√x, we get

$I=x^{3/2}-2\int \sqrt{x}dx$





Hence the integration of root x formula is (2x^3/2)/3, where c is a constant known as an integration constant.

Integral of square root of x 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 sqrt x by using the indefinite integral.

Proof of integral of sqrt x by using definite integral

To compute the integral of sqrt by using a definite integral calculator, 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.

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

Substituting the values of upper and lower bounds.

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

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

How do you integrate square root of x?

To calculate the integral of the square root of x, we can use different integration techniques, such as integration by power rule, integration by parts, and the u-substitution. Mathematically, the integral of x, is calculated as;

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

What is the integration of one by square root x?

The integration of one by square root x is equal to 2√x. It is expressed as;

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

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