Let’s first understand what are Integrals before learning about Definite and Indefinite Integral.

## Integral Calculus

If we know the f’ of a differentiable function in its domain, we can calculate f. In Differential Calculus, we used to refer to f’ as the derivative of the function f. In Integral Calculus, f is the anti-derivative or primal of the function f’. The process of locating anti-derivatives is known as anti-differentiation or integration. It is, as the name suggests, the inverse of differentiation.

Example:

Given: f(x) = x2 .

Derivative of f(x) i.e., f'(x) = 2x = g(x)

if g(x) = 2x, then anti-derivative or integral of g(x) = ∫ g(x) = x2

The integrals are broadly categorised into two types:

Definite Integrals

Let us now discuss both types of integral with definition and properties in detail.

## Definite Integral

On an interval [a, b], the definite integral of a real-valued function f(x) with respect to a real variable x is represented as:

∫_{a}^{b} f(x) dx

Here,

∫ = Integration symbol

a = Lower limit

b = Upper limit

f(x) = Integrand

dx = Integrating agent

As a result, ∫ab f(x) dx is the definite integral of f(x) with respect to dx from a to b.

### Definite Integral Properties

Let’s discuss some essential properties of definite integrals. These will help in the evaluation of the definite integrals.

∫_{a}^{b} f(x) dx = ∫_{a}^{b} f(t) d(t)

∫_{a}^{b} f(x) dx = – ∫_{a}^{b} f(x) dx

∫_{a}^{a} f(x) dx = 0

∫_{a}^{b} f(x) dx = ∫_{a}^{b} f(a + b – x) dx

∫_{0}^{a} f(x) dx = f(a – x) dx

#### Steps for calculating ∫_{a}^{b} f(x) dx

- Calculate the indefinite integral ∫f(x) dx. Let us call this F. (x). It is not necessary to keep integration constant C. This is due to the fact that if we consider F(x) + C instead of F(x), we get

∫ab f(x) dx = [F(x) + C]ab = [F(b) + C] – [F(a) + C] = F(b) + C – F(a) – C = F(b) – F(a)

As a result, the arbitrary constant will not be used to calculate the value of the definite integral.

- Determine the value of F(b) – F(a) = [F(x)]ab

Hence, the value of ∫ab f(x) dx = F(b) – F(a)

## Indefinite Integrals

An indefinite integral is one that does not have any upper or lower bounds.

If F(x) is any anti-derivative of f(x), then the greatest generic antiderivative of f(x) is known as an indefinite integral and is denoted,

∫f(x) dx = F(x) + C

Function antiderivatives and integrals are not unique. There are an endless number of antiderivatives of each function that may be obtained by arbitrarily selecting C from the set of real numbers. Therefore, C is commonly referred to as an arbitrary constant. C is the parameter that determines the various antiderivatives (or integrals) of the given function.

### Some Properties of Indefinite Integral

- ∫cf(x)dx=c∫f(x)dx

Here, c = constant value

We can extract the multiplicative constants from the indefinite integral.

- ∫−f(x)dx=c∫f(x)dx

The indefinite integral is likewise negative due to the negative function.

- ∫f(x)±g(x)dx=∫f(x)dx±∫g(x)dx

It shows the total and difference of the integrals of the functions as the sum and difference of their individual integrals.

#### Indefinite integrals are integrals without limits. The integration technique can be very useful in two ways.

- To find the function whose derivative has been provided.
- To find the area bounded by a function-given curve under specific conditions.