Fundamental Theorem of Calculus, Part 2

$\int_a^b f(x)\,dx=F(b)-F(a)$ for any antiderivative $F$ of $f$ — the evaluation shortcut.

The Fundamental Theorem of Calculus, Part 2 (the Evaluation Theorem) states that if $F$ is any antiderivative of a continuous $f$ , then $\int_a^b f(x)\,dx=F(b)-F(a)$ . This is what makes evaluating a definite integral practical: find ONE antiderivative, evaluate it at the two bounds, subtract — no limit of Riemann sums required.

Examples

$\int_1^3 2x\,dx = \left[x^2\right]_1^3 = 9-1=8$ .