• 54 questions

AP Calc Practice Test on Integration

Unofficial AP Calculus integration practice: Riemann sums, the Fundamental Theorem, antiderivatives, substitution and the BC techniques.

Evaluate the integral from 1 to 3 of (6x² − 4x) dx.

Pick an answer, then press Check answer. Then you see Correct or Not quite, with the reason. Keys: A to D pick, Enter checks.

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Unofficial practice questions, not affiliated with the College Board. Bank checked October 11, 2026.

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All 54 questions

Topics 6.1 to 6.3: Accumulation and Riemann sums 9

  • A table gives f(0) = 2, f(1) = 5, f(2) = 6 and f(3) = 3. What is the trapezoidal sum for the integral from 0 to 3 of f with three equal subintervals? Wikipedia, "Riemann sum", Trapezoidal rule; course topic 6.2
  • A table gives f(0) = 3, f(2) = 7, f(5) = 4 and f(9) = 6. What is the right Riemann sum for the integral from 0 to 9 of f, using the three subintervals of the table? Wikipedia, "Riemann sum", Right rule; course topic 6.2
  • A tank fills at a constant 3 liters per minute for 4 minutes, then at 5 liters per minute for 2 minutes. How much water is added in the 6 minutes? Wikipedia, "Integral"; course topic 6.1
  • f is positive and decreasing on [1, 5]. How does a right Riemann sum compare with the integral from 1 to 5 of f? Wikipedia, "Riemann sum"; course topic 6.2
  • The graph of f is concave up on [0, 6]. How does a trapezoidal sum compare with the integral from 0 to 6 of f? Wikipedia, "Trapezoidal rule"; course topic 6.2
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Topics 6.1 to 6.3: Accumulation and Riemann sums, continued

  • The limit as n → ∞ of the sum from i = 1 to n of (2 + 3i/n)² · (3/n) equals which definite integral? Wikipedia, "Riemann sum", Connection with integration; course topic 6.3
  • r(t) is measured in gallons per hour and t in hours. What are the units of the integral from 0 to 6 of r(t) dt? College Board, AP Calculus AB and BC course description, CHA-4.A.4; course topic 6.1
  • Water drains from a pool so that the volume changes at a constant −40 liters per minute for 15 minutes. What is the accumulated change in volume? College Board, AP Calculus AB and BC course description, CHA-4.A.3; course topic 6.1
  • What is the midpoint Riemann sum for the integral from 0 to 4 of x² dx with two equal subintervals? Wikipedia, "Riemann sum", Midpoint rule; course topic 6.2

Topics 6.4, 6.5 and 6.7: The Fundamental Theorem 11

  • Evaluate the integral from 0 to ln 3 of e^x dx. Wikipedia, "Fundamental theorem of calculus"; course topic 6.7
  • Evaluate the integral from 0 to π/4 of sec² x dx. Wikipedia, "Fundamental theorem of calculus"; course topic 6.7
  • Evaluate the integral from 1 to 3 of (6x² − 4x) dx. Wikipedia, "Fundamental theorem of calculus"; course topic 6.7
  • Evaluate the integral from 1 to 8 of x^(−2/3) dx. Wikipedia, "Fundamental theorem of calculus"; course topic 6.7
  • f′ is continuous, f(1) = 4, and the integral from 1 to 5 of f′(x) dx is 7. What is f(5)? Wikipedia, "Fundamental theorem of calculus"; course topic 6.7
  • The graph of f is the line from (0, 0) to (2, 4), then the horizontal line from (2, 4) to (5, 4). If g(x) is the integral from 0 to x of f(t) dt, what is g(5)? Wikipedia, "Integral"; course topic 6.5
  • Let g(x) be the integral from 0 to x of f(t) dt, where f is continuous and increasing on (1, 4). What must be true of g on (1, 4)? Wikipedia, "Fundamental theorem of calculus"; course topic 6.5
  • Let g(x) be the integral from 0 to x of f(t) dt, where f is continuous, positive on (0, 3) and negative on (3, 5). What does g have at x = 3? Wikipedia, "Fundamental theorem of calculus"; course topic 6.5
  • Let g(x) be the integral from 1 to x of (t³ + 2t) dt. What is g′(x)? Wikipedia, "Fundamental theorem of calculus"; course topic 6.4
  • Let h(x) be the integral from 0 to x² of cos t dt. What is h′(x)? Wikipedia, "Fundamental theorem of calculus"; course topic 6.4
  • Which function is an antiderivative of f(x) = e^(−x²)? Wikipedia, "Antiderivative"; course topics 6.4 and 6.7

Topic 6.6: Properties of definite integrals 8

  • Evaluate the integral from 5 to 5 of (x³ + 7) dx. Wikipedia, "Integral", Properties; course topic 6.6
  • Evaluate the integral from −2 to 3 of |x| dx. Wikipedia, "Integral"; course topic 6.6
  • Evaluate the integral from −3 to 3 of √(9 − x²) dx. Wikipedia, "Integral"; course topic 6.6
  • f(x) = 1 for x < 2 and f(x) = 3 for x ≥ 2. What is the integral from 0 to 5 of f(x) dx? Wikipedia, "Integral"; course topic 6.6
  • The integral from 0 to 3 of f(x) dx is 4. What is the integral from 0 to 3 of (f(x) + 2) dx? Wikipedia, "Integral", Properties; course topic 6.6
  • The integral from 0 to 4 of f is 10 and the integral from 0 to 4 of g is −3. What is the integral from 0 to 4 of (2f(x) − 3g(x)) dx? Wikipedia, "Integral", Properties; course topic 6.6
  • The integral from 1 to 9 of f is 12 and the integral from 1 to 4 of f is 5. What is the integral from 4 to 9 of f? Wikipedia, "Integral", Properties; course topic 6.6
  • The integral from 2 to 7 of f(x) dx is 6. What is the integral from 7 to 2 of f(x) dx? Wikipedia, "Integral", Properties; course topic 6.6

Topic 6.8: Antiderivatives and basic rules 9

  • Find the indefinite integral of (x² + 1)². Wikipedia, "Antiderivative"; course topic 6.8
  • Find the indefinite integral of 1/x³. Wikipedia, "Antiderivative"; course topic 6.8
  • Find the indefinite integral of 1/√(1 − x²). Wikipedia, "Antiderivative"; course topic 6.8
  • Find the indefinite integral of 5/x for x > 0. Wikipedia, "Antiderivative"; course topic 6.8
  • Find the indefinite integral of 8x³ − 6x + 5. Wikipedia, "Antiderivative"; course topic 6.8
  • Find the indefinite integral of sin x + sec² x. Wikipedia, "Antiderivative"; course topic 6.8
  • Find the indefinite integral of √x. Wikipedia, "Antiderivative"; course topic 6.8
  • f′(x) = 6x − 2 and f(1) = 5. What is f(x)? Wikipedia, "Constant of integration"; course topic 6.8
  • Why does an indefinite integral end in + C? Wikipedia, "Constant of integration"; course topic 6.8

Topics 6.9, 6.10 and 6.14: Substitution and rewriting 10

  • Evaluate the integral from 0 to 1 of x e^(x²) dx. Wikipedia, "Integration by substitution"; course topic 6.9
  • Evaluate the integral from 0 to π/2 of sin x cos x dx. Wikipedia, "Integration by substitution"; course topic 6.9
  • Find the indefinite integral of (x² + 3x + 5)/(x + 1) for x > −1. Wikipedia, "Polynomial long division"; course topic 6.10
  • Find the indefinite integral of 1/(x² + 4x + 5). Wikipedia, "Completing the square"; course topic 6.10
  • Find the indefinite integral of 2x cos(x²). Wikipedia, "Integration by substitution"; course topic 6.9
  • Find the indefinite integral of tan x where cos x > 0. Wikipedia, "Integration by substitution"; course topic 6.9
  • Find the indefinite integral of x/(x² + 1). Wikipedia, "Integration by substitution"; course topic 6.9
  • Find the indefinite integral of x²(x³ + 4)⁵. Wikipedia, "Integration by substitution"; course topic 6.9
  • The substitution u = 2x³ + 1 is used on the integral from 0 to 1 of 6x²(2x³ + 1)⁴ dx. Which integral results? Wikipedia, "Integration by substitution"; course topic 6.9
  • Which first step fits the integral of (x³ + 1)/(x − 1)? Wikipedia, "Polynomial long division"; course topics 6.10 and 6.14

Topics 6.11 to 6.13: BC only 7

  • Evaluate the improper integral from 0 to 1 of 1/√x dx. Wikipedia, "Improper integral"; course topic 6.13 (BC only)
  • Evaluate the improper integral from 0 to ∞ of e^(−2x) dx. Wikipedia, "Improper integral"; course topic 6.13 (BC only)
  • Find the indefinite integral of 1/((x − 1)(x + 2)) for x > 1. Wikipedia, "Partial fraction decomposition"; course topic 6.12 (BC only)
  • Find the indefinite integral of ln x. Wikipedia, "Integration by parts"; course topic 6.11 (BC only)
  • Find the indefinite integral of x cos x. Wikipedia, "Integration by parts"; course topic 6.11 (BC only)
  • In the decomposition (5x + 1)/((x − 1)(x + 2)) = A/(x − 1) + B/(x + 2), what are A and B? Wikipedia, "Partial fraction decomposition"; course topic 6.12 (BC only)
  • What is the value of the improper integral from 1 to ∞ of 1/x dx? Wikipedia, "Improper integral"; course topic 6.13 (BC only)

How this list was compiled: Our own questions, written to Unit 6 (Integration and Accumulation of Change, topics 6.1 to 6.14) of the College Board's AP Calculus AB and BC Course and Exam Description, effective fall 2020, https://apcentral.collegeboard.org/media/pdf/ap-calculus-ab-and-bc-course-and-exam-description.pdf, read 2026-10-11. Every answer is mathematics worked from what the question prints, and each one was worked out a second time on 2026-10-11, the wrong choices included. The rule behind each answer was read on 2026-10-11 in the Wikipedia article named under the question. Licence: Own work.

How it works

This AP Calc practice test on integration has 54 questions of our own on Unit 6, Integration and Accumulation of Change, for AB and BC. It is unofficial and not affiliated with or endorsed by the College Board.

Read the full guide

What the test covers

Unit 6 has fourteen topics in the College Board's course description, read October 11, 2026, and the bank follows them in six sets: the Fundamental Theorem and accumulation functions (11 questions), antiderivatives (9), substitution, long division and completing the square (10), properties of definite integrals (8), accumulation and Riemann sums (9), and the BC-only topics of integration by parts, partial fractions and improper integrals (7). The split is ours.

How answers were checked

Every answer was worked a second time, and so was every wrong choice. The rule behind each one links to the Wikipedia article it was read in.

  • After any antiderivative, differentiate your answer. If the integrand does not come back, the answer is wrong.
  • In a definite integral by substitution, change the limits when you change the variable.

Questions and answers

Is integration Unit 4 or Unit 6 in AP Calculus?

Unit 6, Integration and Accumulation of Change, in the course description in force since fall 2020. Unit 4 there is Contextual Applications of Differentiation. A class that calls integration its fourth unit follows its own order, and these questions still fit.

How much of the AP Calculus exam is Unit 6?

Between 15 and 20% of the multiple-choice section in both AB and BC, by the College Board's course description. In AB only Unit 5 carries as much.

Is this practice test for AP Calculus AB or BC?

Both. Forty-seven questions are on topics the two courses share. Seven, marked BC only, come last and cover integration by parts, partial fractions and improper integrals, which the course description gives to BC alone.

Does the test include u-substitution?

Yes. Seven questions use substitution, on indefinite integrals and on definite ones where the limits change with the variable. Three more rewrite the integrand first by long division or completing the square.

Are these real AP Calculus exam questions?

No. The College Board owns the real questions, and none appears here even in other words. Every question on this page was written for it from the unit's topic statements.

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