What is an integration calculator?

The Advanced Integral Calculator is the most comprehensive integral solution on the web, allowing you to perform a wide range of integration operations. The only thing you must enter is a function, variable, and bounds.

You can learn the concepts of calculating integrals quickly and easily using an integration calculator with steps. An integral calculator with steps can be used online to evaluate the integral.

Integration calculators are a mathematical tool that provides the facility to make it easy to evaluate the integral. It is a 100% free online tool that helps us to make our work accurate within seconds. This online tool is specially designed to make math easy. 

The integration calculator allows you to learn the step-by-step concept of calculating the integral. You can evaluate the integral with the help of our online integration calculator. 

What is Integration?

Integration is a process of finding the anti-derivatives. It helps to find the central points and area of curves and volume. It also calculates the area under the curve. An online integration calculator is based on an AI technique that is very powerful.

A discrete data set is integrated by summing its discrete components. A collection of small data cannot be measured singularly, so the integral is calculated to find the functions describing the area, displacement, and volume. When algebra and geometry are combined in calculus, the concept of limit is used. In graphs, limits help us study the result of points getting closer to one another until they are almost at the same distance.

Mathematical Definition:

Mathematicians use integration to find a function g(x) whose derivative, Dg(x), is equal to a given function f(x).

 An example of this is f(x), which is usually called the indefinite integral of f. d(x) represents an infinitesimal displacement along x; thus f(x) is equal to d(x). An integral with a definite value, written as

ba f(x)dx

 In this formula, g(a) + g(b) = Dg(x), where Dg(x) = f(x).

How to Solve Integration?

The first thing you must understand before solving for a definite integral is that definite integrals have start and end points, also known. Integrals can be generalized based on functions and domains through which they are integrated. You can evaluate integrals digitally with integration by parts calculator with steps. Among other things, a line integral is expressed with functions of two or more variables. Alternatively, Surface integrals replace curves in 3-dimensional space with characters.

Integral (definite) can be expressed as follows:

ba f(x)dx

How to Use an Integration Calculator? 

Must follow these steps to use our online integration calculator.

  • In 1st step, Enter the value in the equation box.
  • Select the variable.
  • Now, select the upper bound value.
  • Then, select the lower bound value.
  • The output is shown below.

Integration Rules of Basic Functions

Different types of functions have different integration rules. Let us learn here the basic rules for integration of some common functions, such as:

  • Constant
  • Variable
  • Square
  • Reciprocal
  • Exponential
  • Trigonometry

Integration of Constant

By integrating constant function ‘a’, we get:

∫a dx = ax + C

Integration of Variable

If x is any variable, then;

∫x dx = x22 + C

Integration of Square

This is the case if the function given is a square term;

∫x2 dx =  x33

Integration of Reciprocal 

If  1x  is a reciprocal function of x, then the integration of this function is:

∫(1x ) dx = ln|x| + C                (Natural log of x)

Integration of Exponential Function

Using the following rules:

  • ∫ex dx = ex + C
  • ∫ax dx = axln(a)  + C
  • ∫ln(x) dx = x ln(x) − x + C

Integration of Trigonometric Function

  • ∫cos(x) dx = sin(x) + C
  • ∫sin(x) dx = -cos(x) + C
  • ∫sec2(x) dx = tan(x) + C

An Important Integration Fundamental Rules

The important rules for integration are:

  • Power Rule
  • Sum Rule
  • Different Rule
  • Multiplication by Constant
  • Product Rule

Here are the fundamental rules of integration. Here are a few details we should consider.

Power Rule:

The Power Rule states that if , if we integrate x raised to the power n, then;

∫xn dx = (xn+1/n+1) + C

Sum Rule:

 The integration process can be distributed over addition. i.e., 

dx [ f + g ]= fdx + g dx

Difference Rule:  

The integration process can be distributed over differences. i.e., 

dx [ f – g ]= fdx – g dx

Product Rule:

The product rule of integration states that if a function is a product of two functions, then its integration is the integration of the second function multiplied by the first function added to the integration of the first function multiplied by the second function. ∫cf(x) dx = c∫f(x) dx

Multiplication by Constant

Multiplying a function by a constant result in the integration as follows:

∫cf(x) dx = c∫f(x) dx

            As well as the above-mentioned rules, there are two additional integration rules:

Integration by parts

Also known as the product rule of integration, this rule applies to products.

 When two functions are multiplied together, it is a special type of integration. Parts integration follows the following rule:

            ∫ (u)( v) da = u∫ (v) da – ∫ u'(∫ (v) da) da


  1. u is the function of u(a)
  2. v is the function of v(a)
  3. u’ is the derivative of u(a)

Integration by Substitution

           A method to find integrals by substitution is often referred to as the “The Reverse Chain Rule” or the     “U-Substitution Method”.

           The integral is written in the following form as the first step in this method:

             ∫ f(g(x))g'(x)dx