An integral calculator is a tool used to calculate the definite or indefinite integral of a function. It is used in calculus to find the area under a curve or the total accumulated change in a system.

**Enter Information**

Enter two numbers to find their greatest common divisor:

Solution of Example : integral(12, 18) = integral(18, 12) = integral(12, 6) = integral(6, 0) = 6

## What is integration?

Integration is a mathematical operation that involves finding the integral of a function. The integral of a function is the area under the curve of the function between two points, which can be expressed as a definite integral, or the antiderivative of the function, which can be expressed as an indefinite integral.

In simpler terms, integration is a method used to calculate the area under a curve or the amount of space between a curve and the x-axis. This method is widely used in calculus and is a fundamental tool for solving many problems in physics, engineering, economics, and other fields.

Integration is the inverse operation of differentiation, which is a method used to find the rate of change of a function. Together, integration and differentiation form the basis of calculus, which is a branch of mathematics that deals with the study of change and motion.

## How to perform integration (Antiderivation)?

To perform integration, also known as antiderivation, you need to follow these steps:

- Identify the function to be integrated and determine the limits of integration.
- Use integration rules or techniques to find the antiderivative of the function.
- Substitute the limits of integration into the antiderivative and evaluate the integral.

Here are some common integration rules and techniques:

- Power Rule: For any function f(x) = x^n, the antiderivative is F(x) = (x^(n+1))/(n+1) + C, where C is a constant of integration.
- Trigonometric Rule: For functions like sin(x), cos(x), or tan(x), the antiderivative can be found using trigonometric identities.
- Substitution Rule: For complex functions, you can substitute a variable with a simpler expression to make the integration easier.
- Integration by Parts: This is a technique that involves selecting two parts of the integrand and rewriting it in a way that makes integration easier.
- Partial Fractions: This is a method used to integrate a rational function (a function with a polynomial in the numerator and denominator).

After finding the antiderivative using one of these techniques, you can substitute the limits of integration and evaluate the integral. The result will be a numerical value that represents the area under the curve of the function between the given limits of integration.

## Example

Let’s say you want to find the integral of the function f(x) = x^2 between the limits of 0 and 2.

- Identify the function and determine the limits of integration: f(x) = x^2, limits of integration are 0 and 2.
- Use the power rule to find the antiderivative: ∫x^2 dx = (x^3)/3 + C, where C is a constant of integration.
- Substitute the limits of integration and evaluate the integral: ∫(0 to 2) x^2 dx = [(2^3)/3 + C] – [(0^3)/3 + C] = (8/3 + C) – (0/3 + C) = 8/3

Therefore, the area under the curve of the function f(x) = x^2 between the limits of 0 and 2 is 8/3.