Fundamental Theorem of Calculus

The pair of related results that link differentiation and definite integration: (1) if f is integrable on [a,b] and F(x)=∫_a^x f(t) dt then F is an antiderivative of f where f is continuous; (2) if F is any antiderivative of an integrable function f on [a,b], then ∫_a^b f(x) dx = F(b)−F(a).

A pair of results connecting differentiation and integration: (1) if f is integrable on [a,b] and F(x)=∫_a^x f(t) dt then F is continuous and F'(x)=f(x) almost everywhere under mild conditions; (2) if F is any antiderivative of f on [a,b] then ∫_a^b f(x) dx = F(b)-F(a).