1. Concept of Integrals and types of integral

Chaper 7

Integrals

Integration is the act of bringing together smaller components into a single system that functions as one.

Differentiation is the process of finding the derivative of the functions and integration is the process of finding the antiderivative of a function. So, these processes are inverse of each other. So we can say that integration is the inverse process of differentiation or vice versa. The integration is also called the anti-differentiation. In this process, we are provided with the derivative of a function and asked to find out the function (i.e., primitive). 

To represent the antiderivative of “f”, the integral symbol “∫” symbol is introduced. The antiderivative of the function is represented as ∫ f(x) dx. This can also be read as the indefinite integral of the function “f” with respect to x.

Therefore, the symbolic representation of the antiderivative of a function (Integration) is:

y = ∫ f(x) dx

y=∫ f(x) dx = F(x) + C.
Types of Integrals

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:

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.

Properties Indefinite Integral:

1.

2.   or

3. For a finite number of functions f₁, f₂…. f_n and the real numbers p₁, p₂…p_n,

∫[p₁f₁(x) + p₂f₂(x)….+p_nf_n(x) ]dx = p₁∫f₁(x)dx +  p₂∫f₂(x)dx + ….. +  p_n∫f_n(x)dx