Study About Types of Integrals Like Definite and Indefinite Integrals Here!

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

Indefinite 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.