Fundamental theorem of calculus II

A Riemann integral can be computed from any antiderivative.

Fundamental theorem of calculus II

Fundamental theorem of calculus II: Let $a<b$ and let $f:[a,b]\to\mathbb{R}$ be a Riemann integrable function . Suppose $F:[a,b]\to\mathbb{R}$ is continuous on $[a,b]$ , differentiable on $(a,b)$ , and satisfies $F'(x)=f(x)$ for all $x\in(a,b)$ . Then

\int_a^b f(x)\,dx = F(b)-F(a).

Combined with fundamental theorem of calculus I , this explains why differentiation rules can be used to evaluate definite integrals via antiderivatives.