Fundamental Theorem of Calculus

Calculus

principle

Also known as: FTC

Grade 9-12

The theorem stating that differentiation and integration are inverse operations, linking antiderivatives to definite integrals. The central theorem of calculus—unifies its two main operations.

This concept is covered in depth in our integration techniques guide, with worked examples, practice problems, and common mistakes.

Definition

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

💡 Intuition

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

🎯 Core Idea

Part 1: Derivative of integral = original function. Part 2: \int_a^b f'(x) \, dx = f(b) - f(a).

Example

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

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.

Notation

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

🌟 Why It Matters

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

💭 Hint When Stuck

Write down which part of FTC applies, then check whether the upper limit is just x or a function of x.

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).

Related Concepts

Derivative Integral

🚧 Common Stuck Point

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

⚠️ 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.

Go Deeper

Worked Examples

Step-by-step solved problems

Practice Problems

Test your understanding

Formula Explained

Notation, derivation, and common mistakes

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.

Frequently Asked Questions

What is Fundamental Theorem of Calculus in Math?

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

Why is Fundamental Theorem of Calculus important?

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

What do students usually get wrong about Fundamental Theorem of Calculus?

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

What should I learn before Fundamental Theorem of Calculus?

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

Prerequisites

derivative integral

Cross-Subject Connections

area — FTC shows differentiation and area computation are inverse processes continuous function — FTC Part 1 requires the integrand to be continuous

How Fundamental Theorem of Calculus Connects to Other Ideas

To understand fundamental theorem of calculus, you should first be comfortable with derivative and integral.

Want the Full Guide?

This concept is explained step by step in our complete guide:

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

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