What is the difference between definite and indefinite integral?

The main difference between definite and indefinite integrals lies in their purpose, results, and interpretation.

Definite Integral:

A definite integral is an integral that is evaluated over a specific interval, say from a to b.
The result of a definite integral is a real number that represents the area under the curve of the function within the given interval.
It is written as:
∫ from a to b f(x) dx
The definite integral takes into account the limits of integration (a and b), and the output is a numerical value that signifies the accumulated quantity, such as area or total distance.
The fundamental theorem of calculus connects the definite integral with the concept of anti-differentiation. Specifically, if F is an antiderivative of f, then: ∫ from a to b f(x) dx = F(b) - F(a)
This gives the net area between the curve and the x-axis, considering whether the function is above or below the axis.
Indefinite Integral:

An indefinite integral, on the other hand, is an integral without specific limits. It represents the general form of the antiderivative of a function.
The result of an indefinite integral is a function (or family of functions) rather than a specific number.
It is written as:
∫ f(x) dx
The output is the general formula for the antiderivative, including a constant of integration (C) because the process of integration can yield a family of functions, all differing by a constant.
An indefinite integral represents the process of finding a function F such that F'(x) = f(x).
In summary:

Definite integral: Gives a numerical value (area, total accumulated quantity) over a specific interval.
Indefinite integral: Gives a function that represents the family of antiderivatives, with an added constant of integration.