# Integral Calculator

An Integral Calculator solves the definite and indefinite integral values. In mathematics, integration plays a vital role and deals with different shapes and solids to find the area, volume, length.

## What is an Integral Calculator?

An integral calculator is an online tool that helps to find the value of the integration for a given function. It helps you to calculate the value of the integrations in a few seconds. To use this * integral calculator*, enter the function and limit values in the given input boxes.

### Integral Calculator

## How to Use Integral Calculator?

Please follow the steps below to find the value of integration using an online integral calculator:

- Step 1: Go to Cuemath’s online integral calculator.
- Step 2: Choose definite or indefinite integral from a drop-down list to calculate according to the requirement.
- Step 3: Enter the function with respect to x and the limit values in the given input boxes of the integral calculator.
- Step 4: Click on the "Calculate" button to find the value of the integrations for a given function.
- Step 5: Click on the "Reset" button to clear the fields and enter the different functions.

## How Integral Calculator Works?

Integration is defined as the reverse process of differentiation. The integration is represented by ' ∫ '. Derivatives are defined as finding the rate of change of a function with respect to other variables. It deals with the variables such as x and y, functions f(x), and the corresponding changes in the variables x and y. The derivative of a function is represented by f '(x).

The integrals are classified into 2 types: 1. Indefinite integral 2. Definite integral

Indefinite integrals: The indefinite integrals do not have any upper and lower limits. It is represented as ∫f(x)dx

Definite integrals: The definite integrals have upper and lower limits. It is represented as a ∫b f(x) dx

To calculate the area under a curve y = f(x) from x = a to x = b,the fundamental theorem of calculus tells us to first solve the integration g(x) of f(x) by using:

g(x) = ∫ f(x)dx

and then evaluate g(b) − g(a). That is, the area under the curve f(x) from x=a to x=b is _{a }∫^{b} f(x)dx = g(b) - g(a)

There are common functions and rules we follow to find the integration.

## Solved Examples on Integration

**Example 1:** Find the integration value of 5x3+ 2x2 and verify it using the integral calculator.

**Solution:**

= ∫( 5x3+ 2x2)

= ∫(5x3)+ ∫(2x2)

Using multiplication by constant and power rule,

= [5 × (x3+ 1/ 3+ 1)]+ [2 × x2+ 1/ 2 + 1]

= 5x4/ 4 + 2x3/ 3

**Example 2:** Find the integration value of 2 ∫3 (x+3) dx.

**Solution:**

\(= \int\limits_2^3 {x dx} + \int\limits_2^3 {3dx}\). _{2} ∫^{3}xdx + _{2} ∫^{3} 3dx

\(= \frac{x^2}{2}]_2^3 + 3 x]_2^3\)

\(=\frac{1}{2} ( 3^2 - 2^2) + 3(3 - 2)\)

= \(\frac{1}{2}(5) + 3\)

\(=\frac{11}{2}\)

Similarly, you can use the integral calculator to find the value of integrals for the following:

- x3 / 2 for limits x = 2 to x = 5
- 4x2 + 6x