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# Integral Calculator

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.

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Enter two numbers to find their greatest common divisor:

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Fill the calculator form and click on Calculate button to get result here
Calculation 2
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:

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

Here are some common integration rules and techniques:

1. 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.
2. Trigonometric Rule: For functions like sin(x), cos(x), or tan(x), the antiderivative can be found using trigonometric identities.
3. Substitution Rule: For complex functions, you can substitute a variable with a simpler expression to make the integration easier.
4. 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.
5. 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.

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