Fundamental Theorem of Calculus Formula

The Formula

Part 1: \frac{d}{dx}\int_a^x f(t)\,dt = f(x). Part 2: \int_a^b f(x)\,dx = F(b) - F(a) where F' = f.

When to use: Integration undoes differentiation. They're two sides of the same coin.

Quick Example

\frac{d}{dx}\left[\int_0^x f(t) \, dt\right] = f(x) The derivative of the integral is the original function.

Notation

FTC Part 1 and FTC Part 2. F denotes any antiderivative of f, i.e., F'(x) = f(x).

What This Formula Means

The theorem stating that differentiation and integration are inverse operations, linking antiderivatives to definite integrals.

Integration undoes differentiation. They're two sides of the same coin.

Formal View

Part 1: If f is continuous on [a, b] and G(x) = \int_a^x f(t)\,dt, then G'(x) = f(x) for all x \in (a, b). Part 2: If f is continuous on [a, b] and F' = f, then \int_a^b f(x)\,dx = F(b) - F(a).

Worked Examples

Example 1

easy
Let G(x) = \int_0^x (t^2 + 1)\,dt. Find G'(x) using FTC Part 1.

Solution

  1. 1
    FTC Part 1 states: if G(x) = \int_a^x f(t)\,dt, then G'(x) = f(x).
  2. 2
    Here f(t) = t^2 + 1, so G'(x) = f(x) = x^2 + 1.
  3. 3
    No integration is needed — the derivative of an integral with variable upper limit is just the integrand evaluated at x.

Answer

G'(x) = x^2 + 1
FTC Part 1 says differentiation undoes integration when the upper limit is the variable. You simply replace t with x in the integrand. This is the key link showing derivatives and integrals are inverse operations.

Example 2

hard
Let H(x) = \int_1^{x^2} \cos t\,dt. Find H'(x).

Common Mistakes

  • Confusing Part 1 and Part 2 of the FTC: Part 1 says \frac{d}{dx}\int_a^x f(t)\,dt = f(x); Part 2 says \int_a^b f(x)\,dx = F(b) - F(a). They are related but different statements.
  • Forgetting the chain rule in Part 1 when the upper limit is a function: \frac{d}{dx}\int_0^{x^2} f(t)\,dt = f(x^2) \cdot 2x, not just f(x^2).
  • Applying Part 2 when f is not continuous on [a, b] — if f has a discontinuity in the interval, you must split the integral and handle it as an improper integral.

Why This Formula Matters

The central theorem of calculus—unifies its two main operations.

Frequently Asked Questions

What is the Fundamental Theorem of Calculus formula?

The theorem stating that differentiation and integration are inverse operations, linking antiderivatives to definite integrals.

How do you use the Fundamental Theorem of Calculus formula?

Integration undoes differentiation. They're two sides of the same coin.

What do the symbols mean in the Fundamental Theorem of Calculus formula?

FTC Part 1 and FTC Part 2. F denotes any antiderivative of f, i.e., F'(x) = f(x).

Why is the Fundamental Theorem of Calculus formula important in Math?

The central theorem of calculus—unifies its two main operations.

What do students get wrong about Fundamental Theorem of Calculus?

The FTC is why we can use antiderivatives to compute definite integrals.

What should I learn before the Fundamental Theorem of Calculus formula?

Before studying the Fundamental Theorem of Calculus formula, you should understand: derivative, integral.

Want the Full Guide?

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How to Integrate Rational Functions: Long Division and Partial Fractions →
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Related Concepts

DerivativeIntegral