Calculator of Integral

This calculator computes definite and indefinite integrals of mathematical functions. It provides step-by-step solutions and visualizations to help you understand the integration process.

What is an Integral?

An integral is a mathematical concept that represents the area under a curve or the accumulation of quantities. In calculus, integrals are used to find the area between a curve and the x-axis, the total change, or to solve 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 specified limits, while indefinite integrals find the antiderivative of a function.

Types of Integrals

Definite Integral

A definite integral calculates the exact area under a curve between two specified limits, denoted by the integral symbol with bounds. The formula for a definite integral is:

∫[a,b] f(x) dx = F(b) - F(a)

Where F(x) is the antiderivative of f(x), and a and b are the lower and upper limits of integration, respectively.

Indefinite Integral

An indefinite integral finds the antiderivative of a function, which is a function whose derivative is the original function. The result of an indefinite integral includes a constant of integration, denoted by C. The formula for an indefinite integral is:

∫ f(x) dx = F(x) + C

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

How to Use This Calculator

Select the type of integral you want to calculate (definite or indefinite).
Enter the function you want to integrate in the provided input field. Use standard mathematical notation (e.g., x^2, sin(x), e^x).
If you selected a definite integral, enter the lower and upper limits of integration.
Click the "Calculate" button to compute the integral.
Review the result, which includes the computed integral and a step-by-step solution.
Use the "Reset" button to clear the inputs and start a new calculation.

This calculator supports basic mathematical functions, including polynomials, trigonometric functions, exponential functions, and logarithmic functions.

Formula Used

The calculator uses the following formulas to compute integrals:

Definite Integral: ∫[a,b] f(x) dx = F(b) - F(a) Indefinite Integral: ∫ f(x) dx = F(x) + C

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

Example Calculations

Example 1: Definite Integral

Calculate the definite integral of x^2 from 0 to 1.

∫[0,1] x² dx = (x³/3) evaluated from 0 to 1 = (1³/3) - (0³/3) = 1/3 - 0 = 1/3

The result is 1/3.

Example 2: Indefinite Integral

Find the indefinite integral of sin(x).

∫ sin(x) dx = -cos(x) + C

The result is -cos(x) + C, where C is the constant of integration.

Frequently Asked Questions

What is the difference between a definite and indefinite integral?

A definite integral calculates the exact area under a curve between two specified limits, while an indefinite integral finds the antiderivative of a function, which includes a constant of integration.

What types of functions can this calculator integrate?

This calculator supports basic mathematical functions, including polynomials, trigonometric functions, exponential functions, and logarithmic functions.

How do I enter a function in the calculator?

Use standard mathematical notation when entering a function. For example, enter x^2 for x squared, sin(x) for the sine function, and e^x for the exponential function.

What is the constant of integration in an indefinite integral?

The constant of integration (C) represents the family of antiderivatives for a given function. It accounts for the infinite number of possible solutions to an indefinite integral.