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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

Use u=x^2+1 to integrate 2x(x2+1)3 dx2x(x^2+1)^3\,dx.

Please select an answer first.

Correct answer: A

The correct answer is (x2+1)44+C\frac{(x^2+1)^4}{4}+C.

  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: Since du=2x dx, integrate u^3 to obtain u^4/4.
  3. Check bounds and interpretation: The correct answer is (x2+1)44+C\frac{(x^2+1)^4}{4}+C.

Since du=2x dx, integrate u^3 to obtain u^4/4.

2. QuestionMultiple choice

What substitution is natural for ∫cos⁡(3x) dx\int \cos(3x)\,dx?

Please select an answer first.

Correct answer: B

The correct answer is u=3xu=3x.

  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 inner linear expression 3x is the useful substitution.
  3. Check bounds and interpretation: The correct answer is u=3xu=3x.

The inner linear expression 3x is the useful substitution.

3. QuestionMultiple choice

Evaluate ∫012x(x2+1)2 dx\int_0^1 2x(x^2+1)^2\,dx.

Please select an answer first.

Correct answer: C

The correct answer is 73\frac{7}{3}.

  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: Let u=x^2+1; the bounds become 1 and 2, and ∫1^2 u^2 du=7/3.
  3. Check bounds and interpretation: The correct answer is 73\frac{7}{3}.

Let u=x^2+1; the bounds become 1 and 2, and ∫1^2 u^2 du=7/3.

4. QuestionMultiple choice

What is ∫2xx2+5 dx\int \frac{2x}{x^2+5}\,dx?

Please select an answer first.

Correct answer: D

The correct answer is ln⁡(x2+5)+C\ln(x^2+5)+C.

  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 numerator is the derivative of x^2+5, so the integral is its natural logarithm.
  3. Check bounds and interpretation: The correct answer is ln⁡(x2+5)+C\ln(x^2+5)+C.

The numerator is the derivative of x^2+5, so the integral is its natural logarithm.

5. QuestionMultiple choice

What is the area between y=x and y=x^2 on [0,1]?

Please select an answer first.

Correct answer: A

The correct answer is 16\frac{1}{6}.

  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: On [0,1], x is above x^2; integrate x-x^2 to get 1/2-1/3=1/6.
  3. Check bounds and interpretation: The correct answer is 16\frac{1}{6}.

On [0,1], x is above x^2; integrate x-x^2 to get 1/2-1/3=1/6.

6. QuestionMultiple choice

What is the area between y=4 and y=x^2 on [-2,2]?

Please select an answer first.

Correct answer: B

The correct answer is 323\frac{32}{3}.

  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: Integrate 4-x^2 from -2 to 2.
  3. Check bounds and interpretation: The correct answer is 323\frac{32}{3}.

Integrate 4-x^2 from -2 to 2.

7. QuestionMultiple choice

If velocity is v(t)=3t^2, what is displacement from t=0 to t=2?

Please select an answer first.

Correct answer: C

The correct answer is 8.

  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: Displacement is ∫0^2 3t^2 dt=t^3 evaluated at 2.
  3. Check bounds and interpretation: The correct answer is 8.

Displacement is ∫0^2 3t^2 dt=t^3 evaluated at 2.

8. QuestionMultiple choice

If rate of water flow is r(t)=5-t for 0≤t≤5, how much water enters?

Please select an answer first.

Correct answer: D

The correct answer is 252\frac{25}{2}.

  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: Integrate 5-t from 0 to 5: 25-25/2=25/2.
  3. Check bounds and interpretation: The correct answer is 252\frac{25}{2}.

Integrate 5-t from 0 to 5: 25-25/2=25/2.

9. QuestionMultiple choice

What does the integral of acceleration over a time interval give?

Please select an answer first.

Correct answer: A

The correct answer is Change in velocity.

  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: Integrating acceleration accumulates velocity change.
  3. Check bounds and interpretation: The correct answer is Change in velocity.

Integrating acceleration accumulates velocity change.

10. QuestionMultiple choice

What is the area of a region when the upper curve is f and lower curve is g?

Please select an answer first.

Correct answer: B

The correct answer is ∫ab[f(x)−g(x)] dx\int_a^b [f(x)-g(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: Area uses upper minus lower over the interval.
  3. Check bounds and interpretation: The correct answer is ∫ab[f(x)−g(x)] dx\int_a^b [f(x)-g(x)]\,dx.

Area uses upper minus lower over the interval.

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