What is the Difference Between Definite and Indefinite Integrals?

The main difference between definite and indefinite integrals lies in the context of the limits of integration.

Indefinite Integrals:

• An indefinite integral is a function that, when differentiated, gives the original function (f(x)).
• It has no specified limits of integration, and the constant of integration is arbitrary.
• The process of finding an indefinite integral is also called integration or integrating.

Definite Integrals:

• A definite integral has limits of integration, which are specified as part of the problem.
• The limit values define a specific area under the curve of the function (f(x)).
• To compute a definite integral, one must first find the antiderivative (indefinite integral) of the function and then evaluate it at the endpoints (a and b).

In summary, an indefinite integral represents a function that, when differentiated, gives the original function, while a definite integral represents a specific area under the curve of a function.

Comparative Table: Definite vs Indefinite Integrals

Definite and indefinite integrals are both types of integrals used in calculus, but they serve different purposes and are applied in different contexts. Here is a table highlighting the differences between them:

In summary, definite integration calculates the area under a curve between two specific points (a and b), while indefinite integration finds the general form of a function that, when differentiated, produces a given function. Definite integration requires specifying the lower and upper limits of integration, while indefinite integration does not involve specific intervals but aims to determine a family of functions that differ only by a constant. Both definite and indefinite integration can be represented by the symbol $$\int$$, followed by the function to be integrated and the differential variable.