In AP Calculus, the integrand is the function being integrated, the expression sitting between the integral sign and dx. Reading the integrand tells you which antidifferentiation technique to use and, in improper integrals, whether the integral might be unbounded on the interval.
The integrand is the function inside an integral. In ∫ f(x) dx, the integrand is f(x). It's the thing you're trying to antidifferentiate, or in a definite integral, the thing whose accumulated value you're measuring between the limits of integration.
Think of the integral notation as a sandwich. The integral sign and dx are the bread, and the integrand is the filling. Everything interesting happens in the filling. The shape of the integrand decides which technique works (u-substitution? completing the square? a basic antiderivative rule?), and its behavior decides whether a definite integral is even ordinary. The CED defines an improper integral as one with infinite limits or an integrand that is unbounded somewhere on the interval of integration. So spotting a vertical asymptote in the integrand is just as important as spotting an infinity in the limits.
This term threads through Unit 6 (Integration and Accumulation of Change) and Unit 8 (Applications of Integration). Learning objective AP Calc 6.13.A asks you to evaluate an improper integral or determine that it diverges, and the essential knowledge defines improperness partly by the integrand being unbounded on the interval. Topic 6.14 is essentially a course in integrand-reading. You look at the function's structure and match it to a technique. In Unit 8, learning objective AP Calc 8.9.A has you build volumes with the disc method, where the whole job is constructing the right integrand, π[f(x)]², before you ever compute anything. On the exam, most integration mistakes are really integrand mistakes, either misreading its structure or building the wrong one in an applied setup.
Keep studying AP Calculus Unit 6
Visual cheatsheet
view galleryAntiderivative (Unit 6)
The antiderivative is the answer; the integrand is the question. When you integrate, you're hunting for a function whose derivative equals the integrand. Mixing these two up is the most common vocabulary slip in Unit 6.
Improper Integrals and Divergence (Unit 6)
An integral can be improper for two reasons, and one of them lives entirely in the integrand. If the integrand blows up somewhere on the interval (like 1/√x near x = 0), you have to rewrite the integral as a limit, exactly what 6.13.A tests.
Selecting Techniques for Antidifferentiation (Unit 6)
Topic 6.14 is pattern recognition on the integrand. A function times its derivative's cousin suggests u-substitution; a quadratic under a square root suggests completing the square. The integrand's structure is the only clue you get.
Volume with the Disc Method (Unit 8)
In Unit 8 you don't just read integrands, you build them. Revolving y = f(x) around the x-axis means your integrand is π[f(x)]². Setup points on FRQs are awarded for writing the correct integrand, even if you never finish the arithmetic.
Riemann Sum (Unit 6)
A definite integral is the limit of a Riemann sum, and the integrand is the function whose heights you're sampling in every rectangle. That's why the integrand carries the units of a rate when the integral accumulates a total.
Multiple-choice questions use "integrand" directly in the stem, especially with improper integrals. Expect questions like which property lets you evaluate an improper integral when the integrand contains a removable discontinuity, or how to classify an integral whose integrand is unbounded on the interval. On FRQs, the word matters less than the skill. Released questions like 2021 Q1 (bacteria density in a petri dish) and 2024 Q6 make you construct an integrand from a real situation, such as 2πr·f(r) for density over a circular region or a difference of functions for a region between curves. Graders award setup points for the integrand and limits of integration, so a correct integrand earns credit even if the evaluation goes sideways. Your jobs are to (1) read an integrand and pick a technique, (2) check whether it's unbounded and treat the integral as improper, and (3) build the right integrand in applied contexts like volumes and accumulation.
The integrand is the input; the antiderivative is the output. In ∫ 2x dx = x² + C, the integrand is 2x and the antiderivative is x² + C. If you differentiate the antiderivative, you get the integrand back. On the exam, saying "the antiderivative is unbounded" when you mean the integrand will cost you, since improperness is defined by the integrand's behavior, not the answer's.
The integrand is the function between the integral sign and dx, the thing you are actually integrating.
An integral is improper if its limits are infinite or its integrand is unbounded somewhere on the interval, so always check the integrand for vertical asymptotes inside the limits of integration.
Choosing an antidifferentiation technique in Topic 6.14 starts with reading the integrand's structure, not with memorizing problem types.
In the disc method, you build the integrand yourself as π[radius]², and FRQ setup points reward writing that integrand correctly even before you evaluate.
Don't confuse the integrand with the antiderivative; the integrand is what you start with, and the antiderivative is what you find.
If the integrand has a removable discontinuity on the interval, the integral can still be evaluated using limits of definite integrals.
The integrand is the function being integrated, the expression between the integral sign and dx. In ∫ f(x) dx, the integrand is f(x), and the goal is to find its antiderivative or its accumulated value over an interval.
No. The integrand is the function inside the integral, and the antiderivative is the result of integrating it. In ∫ 2x dx = x² + C, the integrand is 2x and the antiderivative is x² + C.
Yes. The CED says an integral is improper if it has infinite limits or an integrand that is unbounded on the interval of integration. So ∫ from 0 to 1 of 1/√x dx is improper because the integrand blows up at x = 0, even though both limits are finite numbers.
The integrand is the function you're integrating, while the limits of integration are the interval endpoints attached to the integral sign in a definite integral. The integrand determines your technique; the limits determine where you accumulate and whether you might be dealing with a Type I improper integral (one with an infinite limit).
It means the integrand approaches infinity somewhere on the interval, usually at a vertical asymptote. You handle it by rewriting the integral as a limit of definite integrals; if that limit doesn't exist as a finite number, the integral diverges (LO 6.13.A).