Integration - Main Theorem

The Fundamental Theorem of Calculus is the crown jewel of single-variable calculus, establishing the inverse relationship between differentiation and integration. This theorem transforms integration from a limiting process into a computational algorithm.

TheoremFundamental Theorem of Calculus (Part 1)

If $f$ is continuous on $[a, b]$ , then the function $F(x) = \int_a^x f(t)\,dt$ is continuous on $[a, b]$ , differentiable on $(a, b)$ , and $F'(x) = f(x)$ for all $x \in (a, b)$ .

Part 1 states that the integral function $F(x)$ is an antiderivative of $f$ . In other words, differentiation "undoes" integration.

ExampleApplying FTC Part 1

Find $\frac{d}{dx}\int_0^x \sin(t^2)\,dt$ .

By FTC Part 1, with $f(t) = \sin(t^2)$ : $\frac{d}{dx}\int_0^x \sin(t^2)\,dt = \sin(x^2)$

Even though $\sin(t^2)$ has no elementary antiderivative, we can differentiate the integral directly.

ExampleChain Rule with FTC Part 1

Find $\frac{d}{dx}\int_0^{x^2} \sin(t^2)\,dt$ .

Let $u = x^2$ . By the chain rule: $\frac{d}{dx}\int_0^{x^2} \sin(t^2)\,dt = \frac{d}{du}\int_0^u \sin(t^2)\,dt \cdot \frac{du}{dx} = \sin(u^2) \cdot 2x = 2x\sin(x^4)$

TheoremFundamental Theorem of Calculus (Part 2)

If $f$ is continuous on $[a, b]$ and $F$ is any antiderivative of $f$ on $[a, b]$ , then $\int_a^b f(x)\,dx = F(b) - F(a)$

This is commonly written as $\int_a^b f(x)\,dx = \left[F(x)\right]_a^b$ or $\int_a^b f(x)\,dx = F(x)\Big|_a^b$ .

Part 2 provides the evaluation algorithm: to compute a definite integral, find any antiderivative and evaluate it at the endpoints.

ExampleEvaluating Definite Integrals

Compute $\int_1^3 (2x + 1)\,dx$ .

An antiderivative is $F(x) = x^2 + x$ . By FTC Part 2: $\int_1^3 (2x + 1)\,dx = [x^2 + x]_1^3 = (9 + 3) - (1 + 1) = 12 - 2 = 10$

ExampleTrigonometric Integral

Evaluate $\int_0^{\pi/2} \cos x\,dx$ .

$\int_0^{\pi/2} \cos x\,dx = [\sin x]_0^{\pi/2} = \sin(\pi/2) - \sin(0) = 1 - 0 = 1$

Geometrically, this represents the area under one arch of the cosine curve.

Remark

The two parts of the FTC are complementary:

Part 1: Integration followed by differentiation returns the original function
Part 2: Definite integrals can be computed using antiderivatives

Together, they establish that differentiation and integration are inverse operations, unifying the two major branches of calculus.

ExampleNet Change Theorem

If $v(t)$ represents velocity, then by FTC Part 2: $\int_a^b v(t)\,dt = s(b) - s(a)$ where $s(t)$ is position. This is the Net Change Theorem: the integral of a rate of change equals the total change in the quantity.

Similarly:

$\int_a^b a(t)\,dt = v(b) - v(a)$ (acceleration → velocity change)
$\int_a^b C'(x)\,dx = C(b) - C(a)$ (marginal cost → total cost change)

The Fundamental Theorem of Calculus is arguably the most important theorem in all of calculus, providing both theoretical insight and computational power.