In the most commonly used convention (e.g., Apostol 1967, pp. 202-204), the first fundamental theorem of calculus, also termed "the fundamental theorem,
part I" (e.g., Sisson and Szarvas 2016, p. 452) and "the fundmental
theorem of the integral calculus" (e.g., Hardy 1958, p. 322) states that
for
a real-valued continuous function on an open interval and any number in , if is defined by the integral (antiderivative)
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.
a real-valued continuous function on an open interval 
and 
any number in 
, if 
is defined by the integral (antiderivative)
, where 
is the derivative of 
.
