First Fundamental Theorem of Calculus

The first fundamental theorem of calculus states that, if  is continuous 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.

REFERENCES:

Anton, H. "The First Fundamental Theorem of Calculus." §5.8 Calculus: A New Horizon, 6th ed. New York: Wiley, pp. 326-335, 1999.

Apostol, T. M. "The Derivative of an Indefinite Integral. The First Fundamental Theorem of Calculus." §5.1 in Calculus, 2nd ed., Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra. Waltham, MA: Blaisdell, pp. 202-204, 1967.