In the most commonly used convention (e.g., Apostol 1967, pp. 205-207), the second fundamental theorem of calculus, also termed "the fundamental theorem,
part II" (e.g., Sisson and Szarvas 2016, p. 456), states that if is a real-valued continuous
function on the closed interval and is the indefinite integral
of
on ,
then
This result, while taught early in elementary calculus courses, is actually a very deep result connecting the purely algebraic indefinite
integral and the purely analytic (or geometric) definite
integral.
Unfortunately the terminology identifying he "first" and "second" fundamental theorems in sometimes transposed (e.g., Anton 1984), so care is needed identifying the meaning of these appellations when encountered in the wild.