MATH 01130 · Integrals (intro)

Fundamental Theorem of Calculus

Definition

FTC1: if F(x)=∫ₐˣ f(t) dt with f continuous, then F′(x)=f(x). FTC2: ∫ₐᵇ f = F(b)−F(a) for any antiderivative F of f.

FTC links derivatives and integrals as inverse processes.

Watch chain rule with variable limits: d/dx ∫ₐᵍ⁽ˣ⁾ f = f(g(x)) g′(x).

Compute definite integrals by finding any antiderivative and evaluating — no need for the +C.

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