Newton–Leibniz formula

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

Corollary (Newton–Leibniz): $\int_a^b f(x)\,dx = F(b)-F(a).$

This is the standard evaluation rule for definite integrals using antiderivatives .

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