How is the definite integral distinguished from the indefinite integral?

The distinguishing feature of the definite integral is that it involves specific limits of integration, typically denoted by two values (for example, from (a) to (b)). This specification allows for the definite integral to calculate the net area under a curve between these two points on the x-axis. It provides a numerical value that represents this area, which can be positive, negative, or zero, depending on the function and the limits.

In contrast, the indefinite integral does not have these limits and instead represents a family of functions that are antiderivatives of a given function. It includes a constant of integration (usually denoted as (C)), indicating that there are infinitely many antiderivatives for any given function, differing by a constant.

The correct answer highlights this critical distinction in the structure of the two types of integrals, which is fundamental in calculus and particularly significant for applications like calculating areas and understanding functions' behavior over an interval. Understanding this distinction is crucial for effectively applying integral calculus in mathematical problems and engineering contexts.