# Integration

**Integration** is the calculation of an integral. Integrals in maths are used to find many useful quantities such as areas, volumes, displacement, etc. When we speak about integrals, it is related to usually definite integrals. The indefinite integrals are used for anti-derivatives. Integration is one of the two major calculus in Mathematics, apart from differentiation (which measure the rate of change of any function with respect to its variables).

## Integration Definition

The integration denotes the summation of discrete data. The integral is basically calculated to find the functions which will describe the area, displacement, volume, that occurs due to a collection of small data, which cannot be measured singularly. In a broad sense, in calculus, the idea of limit is used where algebra and geometry are implemented. Limits help us study the result of points on a graph get closer to each other until their distance is almost zero. We know that there are two major types of calculus –

- Differential Calculus
- Integral Calculus

## Maths Integration

In Maths, integration is a method of adding or summing up the parts to find the whole. It is a reverse process of differentiation, where we reduce the functions into parts. Basically, this method is used to find the summation under a very large scale. Calculation of small addition problems is an easy task which we can do manually or by using calculators as well. But for big addition problems, where the limits could reach to even infinity, integration methods are used. Integration and differentiation both are an important parts of calculus. The concept level of these topics are very high, hence it is introduced to us at higher secondary classes and then in engineering or higher education.

## Integral Calculus

We are talking about Integral Calculus here. According to Mathematician Bernhard Riemann,

“Integral is based on a limiting procedure which approximates the area of a curvilinear region by breaking the region into thin vertical slabs.” Learn more about Integral calculus here.

Let us now try to understand what does that mean:

- Take an example of a slope of a line in a graph to see what differential calculus is:

In a slope of a line, we can find the slope by using the slope formula. But what if we are given to find an area of a curve? In a curve, the slope of the points varies and it is then we need differential calculus to find the slope of a curve.

You must be familiar with finding out the derivative of a function using the rules of the derivative. Wasn’t it interesting? Now we are going the other way round to find the original function using the rules in Integrating. We are going to use the same rule backwards to find out the original function.

## Integrals in Maths

We have learned until now the concept of integration. You will come across, two types of integrals in maths:

- Definite Integral

- Indefinite Integral

**Definite Integral**

An integral that contains the upper and lower limits then it is a definite integral. On a real line, x is restricted to lie. Riemann Integral is the other name of the Definite Integral.

A definite Integral is represented as:

\(\int_{a}^{b} f(x)dx\)

**Indefinite Integral**

Indefinite integrals are defined without upper and lower limits. It is represented as:

∫f(x)dx = F(x) + C

Where C is any constant and the function f(x) is called the integrand.

## Integration Formulas

Check below the formulas of integral or integration, which are commonly used in higher-level maths calculations. Using these formulas, you can easily solve any problems related to integration.

Also, get some more complete definite integral formulas here.

## Integration Examples

Solve some problems based on integration concept and formulas here.

**Example 1: Find the integral of the function: \(\int_{0}^{3} x^{2}dx\)**

**Solution:**

Given \(\int_{0}^{3} x^{2}dx\)

= \(\left ( \frac{x^{3}}{3} \right )_{0}^{3}\)

\(= \left ( \frac{3^{3}}{3} \right ) – \left ( \frac{0^{3}}{3} \right )\)

= 9

**Example 2: Find the integral of the function: ∫x ^{2} dx**

**Solution:**

Given ∫x^{2} dx

= (x^{3}/3) + C. is the answer.