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Newton–Leibniz formula

If F is an antiderivative of f, then the integral of f equals F(b)-F(a)

Let f:[a,b]→Rf:[a,b]\to\mathbb{R} be and let F:[a,b]→RF:[a,b]\to\mathbb{R} satisfy F′(x)=f(x)F'(x)=f(x) for all x∈(a,b)x\in(a,b).

Corollary (Newton–Leibniz):

∫abf(x) dx=F(b)−F(a).\int_a^b f(x)\,dx = F(b)-F(a).
Remarks

This is the standard evaluation rule for definite integrals using .

Connection to parent theorem: This is the (Part II).