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Quizzes > Mathematics > Advanced Math

Prove Your Mastery with Our AP Calculus Integrals Quiz

Ready for some definite integrals practice? Dive into this integral calculus quiz and ace every question!

Difficulty: Moderate
2-5mins
Learning OutcomesCheat Sheet
Paper art illustration for AP calc integrals quiz promoting testing of integral calculus skills on dark blue background

This AP Calc integrals quiz helps you practice definite and indefinite integrals, from u‑sub and simple antiderivatives to areas and the Fundamental Theorem of Calculus. Work through step‑by‑step problems, get quick feedback, and check gaps before the exam. Then keep going with more integration practice .

What is the indefinite integral of x^3 with respect to x?
x^4/4 + C
x^2/4 + C
3x^2/2 + C
x^4 + C
The power rule for integration states ? x^n dx = x^(n+1)/(n+1) + C. Here n = 3, so the result is x^4/4 + C. This is directly obtained by increasing the exponent by one and dividing by the new exponent. For more details, see .
What is the value of ?5 dx?
5x + C
x + C
0
5 + C
The integral of a constant a with respect to x is ax + C. In this case, a = 5, so ?5 dx = 5x + C. This follows the constant rule for integrals. For more, see .
What is the indefinite integral of sin(x) dx?
-sin(x) + C
-cos(x) + C
cos(x) + C
sin(x) + C
Since the derivative of cos(x) is -sin(x), the integral of sin(x) is -cos(x) + C. This is a standard result in basic trigonometric integration. For more explanation, see .
What is ? e^{2x} dx?
(1/2)e^{2x} + C
e^{2x} + C
2e^{2x} + C
(1/2)x e^{2x} + C
Using u-substitution with u = 2x gives du = 2 dx, so ? e^{2x} dx = (1/2)e^{2x} + C. You divide by the constant coefficient in the exponent. For a step-by-step guide, see .
What is the indefinite integral of x e^{x^2} dx?
e^{x^2} + C
ln|x| e^{x^2} + C
(1/2)e^{x^2} + C
x^2 e^{x^2} + C
Let u = x^2 so that du = 2x dx. Then ? x e^{x^2} dx = (1/2) ? e^u du = (1/2)e^u + C = (1/2)e^{x^2} + C. This is a standard u-substitution problem. More at .
What is the value of the definite integral ?_0^2 3x^2 dx?
4
12
8
6
Integrate 3x^2 to get x^3, then evaluate from 0 to 2: 2^3 - 0 = 8. Definite integration applies the Fundamental Theorem of Calculus. For more practice, see .
What is the indefinite integral of cos^2(x) dx?
sin(x)cos(x) + C
(1 + cos(2x)) + C
x + cos(2x)/2 + C
x/2 + sin(2x)/4 + C
Use the identity cos^2(x) = (1 + cos(2x))/2. Integrating gives ?(1/2) dx + ?(cos(2x)/2) dx = x/2 + sin(2x)/4 + C. This method is known as the half-angle formula. See for details.
Evaluate the definite integral ?_0^1 1/(1 + x^2) dx.
?/4
ln(2)
1/2
?/2
The antiderivative of 1/(1 + x^2) is arctan(x). Evaluating from 0 to 1 gives arctan(1) - arctan(0) = ?/4 - 0 = ?/4. This uses the arctangent integral formula. More at .
What is ? ln(x) dx?
ln^2(x) + C
ln(x)/x + C
x ln(x) - x + C
x ln(x) + C
Use integration by parts with u = ln(x) and dv = dx. Then du = dx/x and v = x, so ?ln(x)dx = x ln(x) - ?x*(1/x)dx = x ln(x) - x + C. See for a walkthrough.
Find ? x^2/(x^3 + 1) dx.
(1/3) ln|x^3 + 1| + C
(1/2) ln|x^3 + 1| + C
ln|x^3 + 1| + C
x/(x^3 + 1) + C
Let u = x^3 + 1, then du = 3x^2 dx. So ? x^2/(x^3 + 1) dx = (1/3) ? du/u = (1/3) ln|u| + C = (1/3) ln|x^3 + 1| + C. See .
What is the value of ?_{-1}^1 ?(1 - x^2) dx?
1
?/4
?
?/2
The integral represents the area of a semicircle of radius 1, which is (1/2)?(1)^2 = ?/2. It follows from the geometric interpretation of ??(1 - x^2) dx. More at .
Evaluate ?_1^e (ln(x))/x dx.
e - 1
1/2
1
ln(e)
Use the fact that ?(ln(x))/x dx = (1/2)(ln(x))^2 + C. Evaluating from 1 to e gives (1/2)(1)^2 - (1/2)(0)^2 = 1/2. For more examples, see .
Compute ?_0^? x^2 e^{-x} dx.
0
2
1
Infinity
This integral equals ?(3) = 2! = 2, where ?(n) = ?_0^? x^{n-1}e^{-x} dx. For n=3, it gives 2. This is a key result from the Gamma function. See .
What is the indefinite integral of sec^3(x) dx?
(1/2)[sec(x)tan(x) - ln|sec(x)+tan(x)|] + C
sec(x) + ln|sec(x)+tan(x)| + C
(1/3)sec(x)tan(x) + C
(1/2)[sec(x)tan(x) + ln|sec(x)+tan(x)|] + C
Use integration by parts with u = sec(x) and dv = sec^2(x)dx. The result is (1/2)[sec(x)tan(x) + ln|sec(x)+tan(x)|] + C. This is a classic result for ?sec^3(x)dx. See .
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Study Outcomes

  1. Apply Integration Techniques -

    Use substitution, integration by parts, and other methods to solve a variety of indefinite integrals with confidence.

  2. Compute Definite Integrals -

    Evaluate definite integrals by applying the Fundamental Theorem of Calculus and interpreting the results in context.

  3. Identify Appropriate Strategies -

    Recognize when to use substitution, integration by parts, or partial fractions for different integral calculus problems.

  4. Interpret Areas Under Curves -

    Relate definite integrals to the geometric concept of area and apply this understanding to calculate exact values.

  5. Analyze Integral Expressions -

    Break down complex integral problems into manageable steps and choose efficient solution paths.

  6. Develop Problem-Solving Skills -

    Strengthen your approach to ap calc integrals by practicing a variety of definite and indefinite integrals tailored to AP exam standards.

Cheat Sheet

  1. Fundamental Theorem of Calculus -

    The Fundamental Theorem of Calculus bridges differentiation and integration by stating that if F′(x)=f(x), then ∫_a^b f(x)dx = F(b)−F(a). Understanding this theorem (as highlighted by MIT OpenCourseWare) ensures you can evaluate definite integrals efficiently by finding antiderivatives.

  2. Power Rule for Antiderivatives -

    For any real n ≠ −1, ∫ x^n dx = x^(n+1)/(n+1) + C (per Stewart Calculus), making it one of the most used formulas in ap calc integrals. Memorize "add one, divide by new power" as a quick mnemonic to recall the process under pressure.

  3. U-Substitution Technique -

    U-substitution simplifies integrals by letting u equal an inner function (e.g., u = x²+1) so that du matches part of the integrand (2x dx). According to Khan Academy, mastering this method lets you transform complex integrals like ∫2x(x²+1)^5 dx into ∫u^5 du in seconds.

  4. Integration by Parts -

    Based on the product rule, ∫u dv = uv−∫v du (as detailed by University of California textbooks) helps tackle products of functions like ∫x e^x dx. A handy tip is the "LIATE" hierarchy (Logarithmic, Inverse trig, Algebraic, Trig, Exponential) to choose u for a smoother solution path.

  5. Definite Integrals and Area Interpretation -

    Definite integrals compute net signed area under a curve between limits a and b, often visualized as positive area above the x-axis minus area below. Exploring practice problems on integral calculus quiz sites solidifies your understanding of area properties and special cases like improper integrals.

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