Fundamental Theorem of Calculus

The theorem linking differentiation and integration as inverse operations, in two parts.

The Fundamental Theorem of Calculus has two parts. Part 1: if $A(x)=\int_a^x f(t)\,dt$ , then $A'(x)=f(x)$ — accumulating area and differentiating undoes each other. Part 2: if $F$ is any antiderivative of $f$ , then $\int_a^b f(x)\,dx = F(b)-F(a)$ — a definite integral is computed from an antiderivative, no Riemann sum required.

Examples

$\displaystyle\int_1^3 2x\,dx = [x^2]_1^3 = 9-1=8$ , using FTC Part 2 with antiderivative $x^2$ .