Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus (FTC) establishes the profound connection between differentiation and integration: they are inverse operations. Part I says integration followed by differentiation recovers the original function. Part II provides a practical method for evaluating definite integrals using antiderivatives. This is the cornerstone of calculus.

Statements

Theorem6.1FTC Part I

If $f : [a, b] \to \mathbb{R}$ is continuous, then the function

$F(x) = \int_a^x f(t) \, dt$

is differentiable on $(a, b)$ with $F'(x) = f(x)$ for all $x \in (a, b)$ .

Theorem6.2FTC Part II

If $f : [a, b] \to \mathbb{R}$ is continuous and $F$ is any antiderivative of $f$ (i.e., $F' = f$ ), then

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

RemarkRelationship

Part I constructs an antiderivative $F(x) = \int_a^x f(t) \, dt$ for continuous $f$ . Part II uses any antiderivative to evaluate integrals. Together, they say: integration and differentiation are inverse operations.

Applications

ExampleEvaluating definite integrals

To compute $\int_0^2 (3x^2 + 2x) \, dx$ , find an antiderivative: $F(x) = x^3 + x^2$ . By FTC Part II,

$\int_0^2 (3x^2 + 2x) \, dx = F(2) - F(0) = (8 + 4) - 0 = 12.$

ExampleFTC Part I: derivative of an integral

Let $G(x) = \int_0^x \sin(t^2) \, dt$ . By FTC Part I, $G'(x) = \sin(x^2)$ .

ExampleChain rule with FTC

Let $H(x) = \int_0^{x^2} e^{-t^2} \, dt$ . By FTC Part I and the chain rule,

$H'(x) = e^{-(x^2)^2} \cdot 2x = 2x e^{-x^4}.$

Summary

The Fundamental Theorem of Calculus bridges differentiation and integration:

Part I: Differentiation undoes integration.
Part II: Antider ivatives compute definite integrals.
Applications: evaluating integrals, solving differential equations, physics.

See FTC Proof for the proof.