---
title: "Indefinite Integrals — AP Calc Definition & Exam Guide"
description: "An indefinite integral is the family of all antiderivatives of a function, written with + C. Master it for FTC problems, u-sub, and initial value problems."
canonical: "https://fiveable.me/ap-calc/key-terms/indefinite-integrals"
type: "key-term"
subject: "AP Calculus AB/BC"
---

# Indefinite Integrals — AP Calc Definition & Exam Guide

## Definition

An indefinite integral is the general antiderivative of a function, meaning the entire family of functions F(x) + C whose derivative equals the original function f(x). Unlike a definite integral, it has no limits of integration and its answer is a function, not a number.

## What It Is

An [indefinite integral](/ap-calc/key-terms/indefinite-integral "fv-autolink") asks the reverse question of differentiation. Instead of "what's the [derivative](/ap-calc/unit-10-infinite-sequences-and-series-bc-only/finding-taylor-polynomial-approximations-functions/study-guide/LszguYzKz0M6GdqTRSr6 "fv-autolink") of this function?" it asks "what function has THIS as its derivative?" Write it as ∫f(x) dx = F(x) + C, where F'(x) = f(x). That F is called an antiderivative, and the indefinite integral is the whole family of them at once.

The "+ C" isn't decoration. Since the derivative of any constant is zero, x² + 5 and x² − 17 both differentiate to 2x. So 2x doesn't have one antiderivative, it has infinitely many, all vertical shifts of each other. The + C captures every single one. Graders specifically look for it, and dropping it is one of the most common point losses in [AP Calc](/ap-calc "fv-autolink"). The notation also matters more than it looks. The dx tells you which variable you're integrating with respect to, which becomes essential once you hit u-substitution.

## Why It Matters

Indefinite integrals live in [Unit 6](/ap-calc/unit-6 "fv-autolink") (Integration and Accumulation of Change) in both AP Calc AB and BC, where you learn that integration and differentiation are inverse processes. This idea is formalized by the Fundamental Theorem of Calculus, which is arguably the single most important result in the entire course. Every basic [antiderivative](/ap-calc/key-terms/antiderivative "fv-autolink") rule you memorize (power rule in reverse, ∫sin x dx = −cos x + C, ∫eˣ dx = eˣ + C, and so on) is really just a derivative rule read backwards.

The payoff stretches across the back half of the course. You need indefinite integrals to evaluate [definite integrals](/ap-calc/unit-6/approximating-areas-with-riemann-sums/study-guide/juN9YbvFYlJtpsMl "fv-autolink") via the FTC, to solve differential equations and initial value problems in Unit 7, and to set up every application in Unit 8, from average value to volumes of revolution. If your antiderivative skills are shaky, everything after Unit 6 gets harder.

## Connections

### Integration by Substitution (Unit 6)

[U-substitution](/ap-calc/key-terms/u-substitution "fv-autolink") is the chain rule run in reverse, and it's your main tool when an indefinite integral isn't in a basic form. If you can spot a function and its derivative sitting inside the integrand, u-sub turns a messy integral into one from your memorized list.

### Integration by Parts (Unit 6, BC only)

[Integration by parts](/ap-calc/unit-6/integrating-using-integration-by-parts/study-guide/O4P3LchNoZnWElf8zETV "fv-autolink") is the product rule run in reverse. It handles indefinite integrals like ∫x·eˣ dx that substitution can't crack, and it only shows up on the BC exam.

### Initial Value Problems (Unit 7)

An initial value problem is where the + C finally earns its keep. You integrate to get a family of solutions, then plug in a known point like f(0) = 3 to solve for C and pin down the one specific function the problem wants.

### [Candidates Test (Unit 5)](/ap-calc/key-terms/candidates-test)

These two connect through FRQs that give you a graph of f' and ask about f. Finding f means antidifferentiating (often with the FTC), and then the Candidates Test uses those values to locate absolute extrema. It's a classic pairing on the free-response section.

## On the AP Exam

Multiple-choice questions test indefinite integrals directly with stems like "∫(3x² − sin x) dx =" where the answer choices differ by sign errors, coefficient slips, or a missing + C. You'll also see them disguised inside differential equation and motion problems, like recovering a position function from velocity.

On the FRQ side, indefinite integration is rarely a standalone question, but it's embedded everywhere. Differential equation FRQs require you to antidifferentiate, include + C, and then solve for C using an initial condition, and the rubric typically awards a separate point for handling the constant correctly. Motion FRQs make you integrate acceleration to get velocity, or velocity to get position. The skill being graded isn't just finding an antiderivative; it's keeping the + C and using given information to resolve it.

## Indefinite Integrals vs Definite Integrals

An indefinite integral has no bounds and produces a family of functions plus C. A definite integral has bounds (like ∫ from 0 to 4) and produces a single number representing accumulated change or net area. The Fundamental Theorem of Calculus connects them. You evaluate a definite integral by finding an antiderivative and computing F(b) − F(a), and the C cancels in that subtraction, which is exactly why definite integrals don't need one.

## Key Takeaways

- An indefinite integral ∫f(x) dx represents every antiderivative of f(x), written as F(x) + C where F'(x) = f(x).
- Always include + C in an indefinite integral; AP rubrics award (and dock) points for it, especially in differential equation FRQs.
- An indefinite integral gives you a function, while a definite integral with bounds gives you a number; the FTC links the two.
- Every antiderivative rule is a derivative rule in reverse, so the reverse power rule adds 1 to the exponent and divides by the new exponent.
- Initial value problems use a given point, like f(2) = 7, to solve for C and turn the general antiderivative into one specific function.
- U-substitution (chain rule backwards) and, on BC, integration by parts (product rule backwards) extend your basic antiderivative toolkit.

## FAQs

### What is an indefinite integral in AP Calculus?

It's the general antiderivative of a function, written ∫f(x) dx = F(x) + C, where F'(x) = f(x). It represents the entire family of functions whose derivative is f(x), and it appears in Unit 6 of both AB and BC.

### Is an indefinite integral the same as an antiderivative?

Essentially yes, with one nuance. An antiderivative is one specific function F(x) with F'(x) = f(x), while the indefinite integral is the whole family of antiderivatives, which is why it carries the + C. On the AP exam the terms are used almost interchangeably.

### What's the difference between definite and indefinite integrals?

Indefinite integrals have no limits of integration and equal a function plus C. Definite integrals have upper and lower bounds and equal a number (the accumulated change). You compute definite integrals by finding an antiderivative first, then evaluating F(b) − F(a).

### Do I really lose points for forgetting + C on the AP exam?

Yes, on free-response questions you can. Differential equation FRQ rubrics routinely tie a point to including the constant of integration and solving for it with the initial condition. On multiple choice, the wrong answers are often the correct antiderivative without + C, set as a trap.

### Are indefinite integrals on both the AB and BC exams?

Yes. Basic antiderivatives and u-substitution are tested on both AB and BC as part of Unit 6. BC adds extra techniques for finding antiderivatives, including integration by parts and integration using partial fractions.

## Related Study Guides

- [AP Calculus AB Exam Guide](/ap-calc/study-tools/2024-ap-calculus-ab-exam-guide/study-guide/QJuGJ5Vt4ua1EGlCbiiV)

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