Introduction: Why Integrals Are Core to AP Calculus
If there’s one topic that truly defines AP Calculus, it’s integration. Whether you’re taking AP Calculus AB or BC, integrals make up a large portion of both the multiple-choice and free-response sections. Understanding the difference between definite and indefinite integrals — and knowing when to apply each — is critical to scoring a 5.
This guide will walk you through both types of integrals with detailed explanations, worked examples, and exam strategies. By the end, you’ll see how RevisionDojo’s structured AP Calculus resources can help you practice exactly the kind of integral problems the exam expects.
What Are Indefinite Integrals?
An indefinite integral represents the antiderivative of a function. Instead of giving a numerical value, it produces a family of functions.
- General form: ∫ f(x) dx = F(x) + C
- Where F′(x) = f(x), and C is the constant of integration.
Key Properties of Indefinite Integrals
- No limits of integration.
- Always include “+ C” in your final answer.
- Used to find general solutions of differential equations.
Example:
∫ 3x² dx = x³ + C
👉 On the AP exam, missing “+ C” in FRQs can cost you a point.
What Are Definite Integrals?
A definite integral gives the net area under a curve between two bounds a and b. Unlike indefinite integrals, definite integrals produce a number.
- General form: ∫ₐᵇ f(x) dx = F(b) – F(a)
- Where F(x) is any antiderivative of f(x).
Key Properties of Definite Integrals
- Always evaluate using upper – lower limits.
- Can represent area, displacement, or accumulation.
- If the function goes below the x-axis, area contributions are negative.
Example:
∫₀² (2x) dx = [x²]₀² = 4 – 0 = 4
Geometric Interpretation of Integrals
- Indefinite integrals: Represent families of functions (antiderivatives).
- Definite integrals: Represent the signed area under a curve from x = a to x = b.
👉 The AP exam often tests whether students understand this difference conceptually, not just computationally.
The Fundamental Theorem of Calculus (FTC)
This theorem is the bridge between definite and indefinite integrals.
- Part 1 (Differentiation of Integrals):
If F(x) = ∫ₐˣ f(t) dt, then F′(x) = f(x). - Part 2 (Evaluation):
∫ₐᵇ f(x) dx = F(b) – F(a).
👉 Every AP student must know this cold — multiple-choice and FRQs use FTC extensively.
Example Problems
Example 1: Indefinite Integral
∫ (4x³ – 6x² + 2) dx
= x⁴ – 2x³ + 2x + C
Example 2: Definite Integral
∫₁³ (x²) dx = [x³/3]₁³ = (27/3) – (1/3) = 26/3
Example 3: FTC Application
If F(x) = ∫₀ˣ (cos t) dt, find F′(x).
By FTC Part 1, F′(x) = cos(x).
How Integrals Appear on the AP Exam
- Indefinite Integrals
- Compute antiderivatives.
- Solve differential equations.
- Include “+ C” in general solutions.
- Definite Integrals
- Compute exact values.
- Apply FTC.
- Solve accumulation/area problems.
- Word Problems
- Motion (displacement, distance).
- Growth/decay.
- Accumulated change.
👉 RevisionDojo provides categorized practice sets so you can focus on definite vs. indefinite problems separately before mixing them.
Calculator vs. No-Calculator Strategies
- No-Calculator Section: Expect integrals with simple antiderivatives.
- Calculator Section: You may need to evaluate definite integrals numerically.
AP tip: Always set up the integral correctly before reaching for the calculator — points are awarded for setup.
Common Mistakes Students Make
- Forgetting “+ C” in indefinite integrals.
- Mixing up definite and indefinite integrals.
- Forgetting negative contributions when curve dips below the x-axis.
- Misapplying FTC by plugging into f(x) instead of F(x).
Practice Plan: Mastering Integrals
- Week 1: Focus on indefinite integrals (basic antiderivatives).
- Week 2: Move to definite integrals and FTC.
- Week 3: Solve word problems involving accumulation and displacement.
- Week 4: Practice mixed AP past paper questions with both types.
➡️ Use RevisionDojo’s FRQ breakdowns to see exactly how integrals are scored.
Frequently Asked Questions
1. What’s the key difference between definite and indefinite integrals?
Indefinite integrals give a family of functions (antiderivatives). Definite integrals give a single numerical value (area/accumulation).
2. Do I always need “+ C” on indefinite integrals?
Yes, unless the problem specifies an initial condition that lets you solve for C.
3. How do integrals connect to physics problems on the AP exam?
Definite integrals can represent total distance traveled, displacement, or accumulated quantities.
4. Are Riemann sums tested on the AP exam?
Yes, Riemann sums often appear as an introduction to definite integrals.
5. What’s the best way to study integrals for the AP exam?
Practice past paper FRQs. RevisionDojo organizes integrals by type and provides step-by-step solutions.
Conclusion: Unlocking the Power of Integrals
Mastering definite vs indefinite integrals is essential for AP Calculus AB and BC. These problems combine computation, conceptual understanding, and real-world application. By internalizing the Fundamental Theorem of Calculus, practicing with word problems, and using RevisionDojo’s AP Calculus resources, you’ll be ready to handle any integral the exam throws at you — and secure the points needed for a 5.
