Online Calculator for Integrals

What is an Integral Calculator?

An integral calculator is a digital tool designed to compute integrals, which are mathematical operations that find the area under a curve or the antiderivative of a function. Integrals are fundamental in calculus and have applications in physics, engineering, economics, and many other fields.

This online calculator supports both definite and indefinite integrals. Definite integrals calculate the area between a curve and the x-axis over a specific interval, while indefinite integrals find the antiderivative of a function, which is essential for solving differential equations.

How to Use This Calculator

1. Enter the function you want to integrate in the function input field. For example, "x^2" or "sin(x)".
2. For definite integrals, enter the lower and upper limits in the provided fields.
3. Select the type of integral (definite or indefinite) from the dropdown menu.
4. Click the "Calculate" button to compute the integral.
5. View the result, which includes the integral value and a graphical representation of the function and its integral.

The calculator will display the result in a clear format, along with a graph showing the function and its integral. This visual representation helps you understand the relationship between the function and its integral.

Types of Integrals

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

Definite Integrals

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

Where F(x) is the antiderivative of f(x). Definite integrals are used to find areas, volumes, and other quantities that depend on accumulation.

Indefinite Integrals

Indefinite integrals find the antiderivative of a function, which is represented by the integral sign with a differential. The formula for an indefinite integral is:

Where C is the constant of integration. Indefinite integrals are essential for solving differential equations and finding general solutions to problems involving rates of change.

Formula Reference

Here are some common integral formulas that this calculator can compute:

• ∫xⁿ dx = (xⁿ⁺¹)/(n+1) + C (for n ≠ -1)
• ∫eˣ dx = eˣ + C
• ∫sin(x) dx = -cos(x) + C
• ∫cos(x) dx = sin(x) + C
• ∫1/x dx = ln|x| + C
• ∫aˣ dx = (aˣ)/ln(a) + C (for a > 0, a ≠ 1)
• ∫sec(x)tan(x) dx = sec(x) + C
• ∫csc(x)cot(x) dx = -csc(x) + C

These formulas are fundamental in calculus and are used to compute integrals of various functions. The calculator can handle these and many other integral calculations.

Worked Example

Let's compute the definite integral of x² from 0 to 1 using this calculator.

Step 1: Enter the Function

Enter "x^2" in the function input field.

Step 2: Set the Limits

Enter 0 as the lower limit and 1 as the upper limit.

Step 3: Select Integral Type

Choose "Definite Integral" from the dropdown menu.

Step 4: Calculate

Click the "Calculate" button to compute the integral.

Result

The calculator will display the result as 0.3333..., which is the value of the integral of x² from 0 to 1. The graph will show the area under the curve x² between x=0 and x=1.