Integrals Calculator

An integral is a mathematical concept that calculates the area under a curve or the accumulation of quantities. This calculator helps you compute definite and indefinite integrals with precision.

What is an Integral?

Integrals are fundamental in calculus and have applications in physics, engineering, economics, and many other fields. They can represent areas under curves, accumulated quantities, or solutions to differential equations.

There are two main types of integrals: definite integrals and indefinite integrals. Definite integrals calculate the exact area under a curve between two points, while indefinite integrals find the antiderivative of a function.

Types of Integrals

Definite Integrals

Definite integrals calculate the exact area under a curve between two specified limits. The formula for a definite integral is:

Definite Integral Formula

∫_a^b f(x) dx = F(b) - F(a)

Where F(x) is the antiderivative of f(x).

Indefinite Integrals

Indefinite integrals find the antiderivative of a function, which represents the family of functions whose derivative is the original function. The result includes a constant of integration (C).

Indefinite Integral Formula

∫ f(x) dx = F(x) + C

Where F(x) is the antiderivative and C is the constant of integration.

How to Use This Calculator

Our integrals calculator is designed to be user-friendly and accurate. Follow these steps to get your results:

Select the type of integral you want to compute (definite or indefinite).
Enter the function you want to integrate in the function field.
For definite integrals, enter the lower and upper limits.
Click the "Calculate" button to compute the integral.
Review the result and explanation provided.

Note

This calculator supports basic mathematical functions. For complex integrals, you may need more advanced tools or software.

Worked Examples

Example 1: Definite Integral

Compute the definite integral of x² from 0 to 1.

Step	Calculation
1	Find the antiderivative of x²: ∫x² dx = (1/3)x³ + C
2	Evaluate at upper limit (1): (1/3)(1)³ = 1/3
3	Evaluate at lower limit (0): (1/3)(0)³ = 0
4	Subtract: 1/3 - 0 = 1/3

The result is 1/3.

Example 2: Indefinite Integral

Compute the indefinite integral of sin(x).

Step	Calculation
1	Find the antiderivative of sin(x): ∫sin(x) dx = -cos(x) + C

The result is -cos(x) + C.

Frequently Asked Questions

What is the difference between definite and indefinite integrals?

Definite integrals calculate the exact area under a curve between two points, while indefinite integrals find the antiderivative of a function, which represents a family of functions.

Can this calculator handle complex integrals?

This calculator supports basic mathematical functions. For complex integrals, you may need more advanced tools or software.

What is the constant of integration in indefinite integrals?

The constant of integration (C) represents the family of functions that have the same derivative. It accounts for the infinite number of possible antiderivatives.