Online Integral Calculator

What is an Integral Calculator?

An integral calculator is an online tool designed to compute integrals of mathematical functions. Integrals are fundamental in calculus and are used to find areas under curves, volumes of solids, and solutions to differential equations. Our online integral calculator supports both definite and indefinite integrals, making it a versatile tool for students and professionals alike.

Integrals can be challenging to compute manually, especially for complex functions. Our calculator simplifies this process by providing accurate results quickly. Whether you need to find the area under a curve or solve a differential equation, our tool offers the precision and convenience you need.

How to Use the Integral Calculator

Using our online integral calculator is straightforward. Follow these steps to get accurate results:

1. Enter the function you want to integrate in the input field. For example, you might enter "x^2" for the function f(x) = x².
2. Specify the type of integral you need: definite or indefinite.
3. For definite integrals, enter the lower and upper limits of integration.
4. Click the "Calculate" button to compute the integral.
5. Review the result displayed on the screen. The calculator will show the integral value and, if available, a graph of the function.

Our calculator supports a wide range of mathematical functions, including polynomials, trigonometric functions, exponential functions, and more. You can also adjust the limits of integration for definite integrals to explore different scenarios.

Types of Integrals

Integrals can be categorized into two main types: definite and indefinite integrals.

Indefinite Integrals

An indefinite integral represents the antiderivative of a function. It is written as ∫f(x)dx and results in a family of functions that differ by a constant. Indefinite integrals are used to find the general solution to differential equations and to compute areas under curves when the limits are not specified.

Definite Integrals

A definite integral calculates the exact area under a curve between specified limits. It is written as ∫[a to b] f(x)dx and results in a single numerical value. Definite integrals are used to compute areas, volumes, and other quantities in calculus and physics.

Formula Used

The integral calculator uses the fundamental theorem of calculus to compute integrals. For a function f(x), the integral is calculated as follows:

The calculator supports a wide range of functions, including polynomials, trigonometric functions, exponential functions, and more. The results are computed using numerical methods for complex functions and symbolic computation for simpler functions.

Worked Example

Let's compute the definite integral of the function f(x) = x² from x = 0 to x = 2.

1. Identify the function and limits: f(x) = x², lower limit a = 0, upper limit b = 2.
2. Find the antiderivative: The antiderivative of x² is (x³)/3.
3. Apply the definite integral formula: ∫[0 to 2] x²dx = (2³)/3 - (0³)/3 = 8/3 - 0 = 8/3.
4. Interpret the result: The area under the curve of x² from 0 to 2 is 8/3 square units.

Using our online integral calculator, you can verify this result quickly and easily. Simply enter the function and limits, and the calculator will provide the exact value of the integral.