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Integrals Practice Tests

These Grade 12 Math practice tests develop integration skills through antiderivatives, definite integrals, substitution, accumulation functions, average value, area between curves, motion, Riemann sums, and the Fundamental Theorem of Calculus.

Skills covered

  • Finding antiderivatives of polynomial, trigonometric, exponential, logarithmic, and composite functions
  • Evaluating definite integrals and interpreting signed area, total change, and units
  • Using substitution, symmetry, and the Fundamental Theorem of Calculus
  • Calculating average values, areas between curves, and accumulated quantities
  • Applying integrals to velocity, flow, distance, Riemann sums, and real-world rates

How to use these practice tests

First identify whether the integral represents an antiderivative, a signed area, or accumulated change. Then select the right rule, substitution, bounds, and units and check the interpretation.

For more Grade 12 Math practice, return to the Grade 12 Math category.

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Grade 12 Integrals Practice Tests

Practice Grade 12 Math questions about antiderivatives, indefinite and definite integrals, substitution, area, accumulation functions, average value, motion, Riemann sums, and the Fundamental Theorem of Calculus.

Choose the best answer. Identify whether the task asks for an antiderivative, accumulation, area, average value, or total change, then apply the appropriate bounds and rule.

10 Questions
0/10 Questions0 %
Correct 0
Incorrect 0
Unanswered 10
1. QuestionMultiple choice

If A(x)=∫0x(t2+4) dtA(x)=\int_0^x (t^2+4)\,dt, what is A'(x)?

Please select an answer first.

Correct answer: A

The correct answer is x2+4x^2+4.

  1. Identify the integral structure: Determine whether the question asks for an antiderivative, a definite accumulation, area, average value, substitution, or a calculus application.
  2. Apply the integration method: The Fundamental Theorem gives the integrand evaluated at x.
  3. Check bounds and interpretation: The correct answer is x2+4x^2+4.

The Fundamental Theorem gives the integrand evaluated at x.

2. QuestionMultiple choice

If F(x)=∫2xcos⁡t dtF(x)=\int_2^x \cos t\,dt, what is F'(x)?

Please select an answer first.

Correct answer: B

The correct answer is cos⁡x\cos x.

  1. Identify the integral structure: Determine whether the question asks for an antiderivative, a definite accumulation, area, average value, substitution, or a calculus application.
  2. Apply the integration method: Differentiate the accumulation function using the Fundamental Theorem.
  3. Check bounds and interpretation: The correct answer is cos⁡x\cos x.

Differentiate the accumulation function using the Fundamental Theorem.

3. QuestionMultiple choice

What is ∫14f′(x) dx\int_1^4 f'(x)\,dx?

Please select an answer first.

Correct answer: C

The correct answer is f(4)-f(1).

  1. Identify the integral structure: Determine whether the question asks for an antiderivative, a definite accumulation, area, average value, substitution, or a calculus application.
  2. Apply the integration method: The Fundamental Theorem evaluates the antiderivative at the endpoints.
  3. Check bounds and interpretation: The correct answer is f(4)-f(1).

The Fundamental Theorem evaluates the antiderivative at the endpoints.

4. QuestionMultiple choice

If F′(x)=f(x)F'(x)=f(x), then ∫abf(x) dx\int_a^b f(x)\,dx equals:

Please select an answer first.

Correct answer: D

The correct answer is F(b)−F(a)F(b)-F(a).

  1. Identify the integral structure: Determine whether the question asks for an antiderivative, a definite accumulation, area, average value, substitution, or a calculus application.
  2. Apply the integration method: Use the upper endpoint value minus the lower endpoint value.
  3. Check bounds and interpretation: The correct answer is F(b)−F(a)F(b)-F(a).

Use the upper endpoint value minus the lower endpoint value.

5. QuestionMultiple choice

What is the average value of f(x)=x on [2,6]?

Please select an answer first.

Correct answer: A

The correct answer is 4.

  1. Identify the integral structure: Determine whether the question asks for an antiderivative, a definite accumulation, area, average value, substitution, or a calculus application.
  2. Apply the integration method: Average value is (1/(6-2))∫2^6 x dx=4.
  3. Check bounds and interpretation: The correct answer is 4.

Average value is (1/(6-2))∫2^6 x dx=4.

6. QuestionMultiple choice

What is the average value formula on [a,b]?

Please select an answer first.

Correct answer: B

The correct answer is 1b−a∫abf(x) dx\frac{1}{b-a}\int_a^b f(x)\,dx.

  1. Identify the integral structure: Determine whether the question asks for an antiderivative, a definite accumulation, area, average value, substitution, or a calculus application.
  2. Apply the integration method: Total accumulation is divided by interval length.
  3. Check bounds and interpretation: The correct answer is 1b−a∫abf(x) dx\frac{1}{b-a}\int_a^b f(x)\,dx.

Total accumulation is divided by interval length.

7. QuestionMultiple choice

If A′(x)=f(x)A'(x)=f(x) and f(x)>0, what happens to A?

Please select an answer first.

Correct answer: C

The correct answer is A is increasing.

  1. Identify the integral structure: Determine whether the question asks for an antiderivative, a definite accumulation, area, average value, substitution, or a calculus application.
  2. Apply the integration method: A positive derivative means the accumulation function increases.
  3. Check bounds and interpretation: The correct answer is A is increasing.

A positive derivative means the accumulation function increases.

8. QuestionMultiple choice

What is ddx∫1x2t3 dt\frac{d}{dx}\int_1^{x^2} t^3\,dt?

Please select an answer first.

Correct answer: D

The correct answer is 2x72x^7.

  1. Identify the integral structure: Determine whether the question asks for an antiderivative, a definite accumulation, area, average value, substitution, or a calculus application.
  2. Apply the integration method: Use the chain rule: (x^2)^3·2x=2x^7.
  3. Check bounds and interpretation: The correct answer is 2x72x^7.

Use the chain rule: (x^2)^3·2x=2x^7.

9. QuestionMultiple choice

What is ddx∫0sin⁡x(1+t2) dt\frac{d}{dx}\int_0^{\sin x} (1+t^2)\,dt?

Please select an answer first.

Correct answer: A

The correct answer is (1+sin⁡2x)cos⁡x(1+\sin^2 x)\cos x.

  1. Identify the integral structure: Determine whether the question asks for an antiderivative, a definite accumulation, area, average value, substitution, or a calculus application.
  2. Apply the integration method: The integrand is evaluated at sin x and multiplied by the derivative cos x.
  3. Check bounds and interpretation: The correct answer is (1+sin⁡2x)cos⁡x(1+\sin^2 x)\cos x.

The integrand is evaluated at sin x and multiplied by the derivative cos x.

10. QuestionMultiple choice

What does an accumulation function measure?

Please select an answer first.

Correct answer: B

The correct answer is Total change collected from a starting point to x.

  1. Identify the integral structure: Determine whether the question asks for an antiderivative, a definite accumulation, area, average value, substitution, or a calculus application.
  2. Apply the integration method: An integral accumulates signed rates over an interval.
  3. Check bounds and interpretation: The correct answer is Total change collected from a starting point to x.

An integral accumulates signed rates over an interval.

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