The first part of the theorem, sometimes called the first fundamental theorem of calculus, shows that an indefinite integration can be reversed by a differentiation. The first part is also important because it guarantees the existence of antiderivatives for continuous functions.
The second part, sometimes called the second fundamental theorem of calculus, allows one to compute the definite integral of a function by using any one of its infinitely many antiderivatives. This part of the theorem has invaluable practical applications, because it markedly simplifies the computation of definite integrals.
The first published statement and proof of a restricted version of the fundamental theorem was byJames Gregory (1638 - 1675). Isaac Barrow (1630 - 1677) proved a more generalized version of the theorem, while Barrow's student Isaac Newton (1643 - 1727) completed the development of the surrounding mathematical theory. Gottfried Leibniz (1646 - 1716) systematized the knowledge into a calculus for infinitesimal quantities. Wikipedia - Fundamental Theorem of Calculus