Calculator for Integrals

What is an Integral?

An integral represents the area under a curve between two points. It can be calculated as the limit of a sum of areas of rectangles under the curve. Integrals are used in physics, engineering, economics, and many other fields to solve problems involving accumulation, area, and volume.

The integral of a function f(x) with respect to x is written as ∫f(x)dx. The result is called the antiderivative of f(x).

Types of Integrals

There are two main types of integrals:

1. Definite Integrals

Definite integrals calculate the exact area under a curve between two specific limits, a and b. The notation is ∫[a to b] f(x)dx.

2. Indefinite Integrals

Indefinite integrals find the general antiderivative of a function, represented as ∫f(x)dx + C, where C is the constant of integration.

Basic Integral Formulas

Here are some fundamental integral formulas:

• ∫xⁿ dx = (xⁿ⁺¹)/(n+1) + C (for n ≠ -1)
• ∫eˣ dx = eˣ + C
• ∫aˣ dx = (aˣ)/ln(a) + C (for a > 0, a ≠ 1)
• ∫sin(x) dx = -cos(x) + C
• ∫cos(x) dx = sin(x) + C
• ∫sec²(x) dx = tan(x) + C
• ∫csc(x)cot(x) dx = -csc(x) + C
• ∫sec(x)tan(x) dx = sec(x) + C

These formulas are essential for solving many calculus problems and are implemented in this calculator.

How to Use This Calculator

Using our integral calculator is simple:

1. Select the type of integral (definite or indefinite)
2. Enter the function you want to integrate
3. For definite integrals, enter the lower and upper limits
4. Click "Calculate" to get the result
5. View the detailed solution and graph (if available)

Example Calculations

Let's look at some example calculations:

Example 1: Indefinite Integral

Calculate ∫x² dx

Using the formula ∫xⁿ dx = (xⁿ⁺¹)/(n+1) + C:

Example 2: Definite Integral

Calculate ∫[0 to 1] x² dx

First find the antiderivative, then apply the limits:

This calculator will perform these calculations for you automatically.