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Integral Calculator

Integral Calculator is used to integrate a function that could be in the form of either a definite or an indefinite integral. Integration is one of the most fundamental operations of calculus. It is the process of uniting infinitesimal data to find a whole.

## What is an Integral Calculator?

Integral calculator is an online tool that helps to find the value of a given definite or indefinite integral. Integration is the reverse process of differentiation. Thus, when we integrate a function we essentially determine its antiderivative. To use this integral calculator, enter the values in the given input boxes.

### Integral Calculator

## How to Use Integral Calculator?

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

- Step 1: Go to Cuemath’s online integral calculator.
- Step 2: Choose definite or indefinite integral from a drop-down list and enter the values in the input boxes.
- Step 3: Click on the “Calculate” button to find the value of the integral for a given function.
- Step 4: Click on the “Reset” button to clear the fields and enter different values.

## How Does Integral Calculator Work?

Integration can be defined as the process of determining the area under a curve. There are two types of integrals namely, definite integrals and indefinite integrals. Given below are the different methods that can be used to integrate a given function:

- Decomposition method – Using this technique, we can break down the given function into the sum and difference of smaller functions whose integral value is known. The given function can be algebraic, trigonometric, exponential, or a combination of the three.
- Integration by substitution – In this technique, we substitute the variable of integration with a different variable. This helps to simplify the process of solving the integral.
- Integration by partial fractions – Suppose our integrand is expressed in the form of an improper rational function. We can use the concept of partial fractions to convert our integrand into a proper rational function. Finally, we can integrate this to get our answer.
- Integration by parts – Suppose our integrand is in the form of ∫f(x)g(x)dx. To solve this problem using integration by parts, we apply the formula: ∫f(x).g(x) dx = f(x) ∫g(x)dx − ∫ [f′(x) ∫ g(x)dx ]dx.

There are many formulas available for the purpose of solving special integrals.

## Solved Examples on Integral Calculator

Example 1:

Find the integration value of \(\int (x^{3} + x + 7)dx\) and verify it using the integral calculator.

Solution:

Given : \(\int (x^{3} + x + 7)dx\)

= \(\frac{x^{4}}{4} + \frac{x^{2}}{2} + 7x + C\)

Example 2:

Find the integration value of \(\int_{2}^{4}[x^{2}+5]dx\) and verify it using the integral calculator.

Solution:

Given : \(\int_{2}^{4}[x^{2}+5]dx\)

= \(_{2}^{4}[\frac{x^{3}}{3} + 5x]\)

= \([\frac{4^{3}}{3} + 5(4)] – [\frac{2^{3}}{3} + 5(2)]\)

= 28.67

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

- \(\int_{5}^{16} (e^{x} + 15x)dx\)
- \(\int \frac{1}{x^{2} + x}dx\)

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