What is an indefinite integral?

An indefinite integral is indeed defined as an antiderivative function without limits. This concept is essential in calculus, as it represents a family of functions whose derivative corresponds to the integrand. When we find the indefinite integral of a function, we are looking for a general solution that can incorporate any constant, since the derivative of a constant is zero. This is why we add the constant of integration (often denoted as C) to the result.

For example, if we take the indefinite integral of the function ( f(x) = 2x ), the result would be ( F(x) = x^2 + C ). Here, ( F(x) ) is an antiderivative of ( f(x) ), and the presence of ( C ) signifies that there are infinitely many possible antiderivatives differing only by a constant.

In contrast, an integral computed over specified limits truly reflects a definite integral, which provides a numerical value representing the area under the curve between those limits. An approximation of area under a curve refers typically to methods like Riemann sums or trapezoidal rule, rather than the direct computation of an indefinite integral. Saying it defines an area that cannot be computed suggests confusion with the concept of areas represented