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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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1. Test

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

What is ∫8x3 dx\int 8x^3\,dx?

Please select an answer first.

Correct answer: A

The correct answer is 2x4+C2x^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: Increase the power to 4 and divide 8 by 4.
  3. Check bounds and interpretation: The correct answer is 2x4+C2x^4+C.

Increase the power to 4 and divide 8 by 4.

2. QuestionMultiple choice

What is ∫(6x2−4x+1) dx\int (6x^2-4x+1)\,dx?

Please select an answer first.

Correct answer: B

The correct answer is 2x3−2x2+x+C2x^3-2x^2+x+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: Integrate each term using the power rule.
  3. Check bounds and interpretation: The correct answer is 2x3−2x2+x+C2x^3-2x^2+x+C.

Integrate each term using the power rule.

3. QuestionMultiple choice

What is ∫sin⁡x dx\int \sin x\,dx?

Please select an answer first.

Correct answer: C

The correct answer is −cos⁡x+C-\cos x+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 derivative of -cos x is sin x.
  3. Check bounds and interpretation: The correct answer is −cos⁡x+C-\cos x+C.

The derivative of -cos x is sin x.

4. QuestionMultiple choice

What is ∫sec⁡2x dx\int \sec^2 x\,dx?

Please select an answer first.

Correct answer: D

The correct answer is tan⁡x+C\tan x+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 derivative of tan x is secant squared.
  3. Check bounds and interpretation: The correct answer is tan⁡x+C\tan x+C.

The derivative of tan x is secant squared.

5. QuestionMultiple choice

What is ∫5ex dx\int 5e^x\,dx?

Please select an answer first.

Correct answer: A

The correct answer is 5ex+C5e^x+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 constant factor remains and e^x is its own antiderivative.
  3. Check bounds and interpretation: The correct answer is 5ex+C5e^x+C.

The constant factor remains and e^x is its own antiderivative.

6. QuestionMultiple choice

What is ∫3x dx\int \frac{3}{x}\,dx for x>0?

Please select an answer first.

Correct answer: B

The correct answer is 3ln⁡x+C3\ln x+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 antiderivative of 1/x is ln x.
  3. Check bounds and interpretation: The correct answer is 3ln⁡x+C3\ln x+C.

The antiderivative of 1/x is ln x.

7. QuestionMultiple choice

Why does an indefinite integral include +C?

Please select an answer first.

Correct answer: C

The correct answer is Antiderivatives differ by an arbitrary constant.

  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: Differentiating any constant gives zero, so all constants represent valid antiderivatives.
  3. Check bounds and interpretation: The correct answer is Antiderivatives differ by an arbitrary constant.

Differentiating any constant gives zero, so all constants represent valid antiderivatives.

8. QuestionMultiple choice

What is ∫(2x+1)4 dx\int (2x+1)^4\,dx?

Please select an answer first.

Correct answer: D

The correct answer is (2x+1)510+C\frac{(2x+1)^5}{10}+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: Use u=2x+1, du=2dx, then integrate u^4.
  3. Check bounds and interpretation: The correct answer is (2x+1)510+C\frac{(2x+1)^5}{10}+C.

Use u=2x+1, du=2dx, then integrate u^4.

9. QuestionMultiple choice

What is ∫0 dx\int 0\,dx?

Please select an answer first.

Correct answer: A

The correct answer is 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: Every constant function is an antiderivative of zero, so the result is an arbitrary constant.
  3. Check bounds and interpretation: The correct answer is C.

Every constant function is an antiderivative of zero, so the result is an arbitrary constant.

10. QuestionMultiple choice

If F′(x)=f(x)F'(x)=f(x), what is an antiderivative of f(x)?

Please select an answer first.

Correct answer: B

The correct answer is F(x)+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: An antiderivative is any original function plus a constant.
  3. Check bounds and interpretation: The correct answer is F(x)+C.

An antiderivative is any original function plus a constant.

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