Fundamental Theorem of Calculus, Part II

Fundamental Theorem of Calculus (Part II): Let $f:[a,b]\to\mathbb{R}$ be continuous . Suppose $F:[a,b]\to\mathbb{R}$ is differentiable on $(a,b)$, continuous on $[a,b]$, and satisfies $F'(x)=f(x)\quad\text{for all }x\in(a,b).$ Then $\int_a^b f(x)\,dx = F(b)-F(a).$

This is the evaluation rule behind essentially all “antiderivative computations” of definite integrals in calculus.

Proof sketch: Let $G(x)=\int_a^x f(t)\,dt$. By Part I , $G'(x)=f(x)$ for all $x\in(a,b)$. Then $(F-G)'=0$ on $(a,b)$, so $F-G$ is constant on $[a,b]$. Evaluating at $a$ gives $F(x)-G(x)=F(a)$, hence $G(b)=F(b)-F(a)$.