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Integration and the Fundamental Theorem Online Quiz Questions

Use this free practice quiz with 20 questions to review Integration and the Fundamental Theorem, test your knowledge, and prepare for your next test or exam.

20 questions
01
Choose one
1 point

Which expression is an antiderivative of x2x^2?

  1. A

    x3+Cx^3+C

  2. B

    x33+C\frac{x^3}{3}+C

  3. C

    2x+C2x+C

  4. D

    3x2+C3x^2+C

02
Choose one
1 point

Which expression gives the general antiderivative of 1x\frac{1}{x} on an interval that does not contain zero?

  1. A

    x−2+Cx^{-2}+C

  2. B

    1x2+C\frac{1}{x^2}+C

  3. C

    ln⁡∣x∣+C\ln|x|+C

  4. D

    ln⁡x+C\ln x+C only for all nonzero xx

03
Choose one
1 point

Evaluate ∫2x(x2+5)4 dx\int 2x(x^2+5)^4\,dx.

  1. A

    (x2+5)5+C(x^2+5)^5+C

  2. B

    (x2+5)55+C\frac{(x^2+5)^5}{5}+C

  3. C

    (x2+5)44+C\frac{(x^2+5)^4}{4}+C

  4. D

    2(x2+5)5+C2(x^2+5)^5+C

04
Choose one
1 point

For f(x)=x2f(x)=x^2 on [0,2][0,2], what is the left Riemann sum L2L_2 using two equal subintervals?

  1. A

    1

  2. B

    2

  3. C

    3

  4. D

    5

05
Choose all
1 point

Select all statements that correctly describe a definite integral as signed accumulation.

  1. A

    Regions below the xx-axis contribute negatively.

  2. B

    A definite integral always equals total geometric area.

  3. C

    Regions above the xx-axis contribute positively.

  4. D

    Positive and negative contributions cannot cancel.

06
Choose all
1 point

Select all statements that correctly describe the trapezoidal rule for an equally spaced partition.

  1. A

    It uses only midpoint values.

  2. B

    It gives Tn=Ln+Rn2T_n=\frac{L_n+R_n}{2}.

  3. C

    Interior partition values receive coefficient 2 in the expanded formula.

  4. D

    It is defined only when the function has an elementary antiderivative.

07
True or false
1 point

True or false: If F′(x)=f(x)F'(x)=f(x), then F(x)+CF(x)+C is also an antiderivative of f(x)f(x) for every constant CC.

  1. A

    True

  2. B

    False

08
True or false
1 point

True or false: If ff is continuous, then ddx∫xaf(t) dt=f(x)\frac{d}{dx}\int_x^a f(t)\,dt=f(x).

  1. A

    True

  2. B

    False

09
Written response
1 point

What mathematical term names the function f(x)f(x) in ∫f(x) dx\int f(x)\,dx? Enter the single term.

10
Fill in the blank
1 point

Complete the two statements of the Fundamental Theorem of Calculus: If F(x)=∫axf(t) dtF(x)=\int_a^x f(t)\,dt, then . If F′(x)=f(x)F'(x)=f(x) on [a,b][a,b], then ∫abf(x) dx=\int_a^b f(x)\,dx=.

11
Fill in the blank
1 point

A definite integral represents . For a velocity function v(t)v(t), integrating ∣v(t)∣|v(t)| gives .

12
Open ended
1 point

Explain how to evaluate ∫cos⁡(3x) dx\int\cos(3x)\,dx using substitution, and justify every factor in your final answer.

13
Choose one
1 point

Let G(x)=∫1x2cos⁡(t3) dtG(x)=\int_1^{x^2}\cos(t^3)\,dt. Which expression is G′(x)G'(x)?

  1. A

    2xcos⁡(x6)2x\cos(x^6)

  2. B

    cos⁡(x6)\cos(x^6)

  3. C

    2xcos⁡(x2)2x\cos(x^2)

  4. D

    sin⁡(x6)\sin(x^6)

14
Choose one
1 point

Which expression is the general antiderivative of 4x34x^3?

  1. A

    x4+Cx^4+C

  2. B

    4x4+C4x^4+C

  3. C

    x3+Cx^3+C

  4. D

    x4x^4

15
Choose one
1 point

For equally spaced subintervals, how is the trapezoidal estimate TnT_n related to the left and right Riemann sums?

  1. A

    Tn=Ln+RnT_n=L_n+R_n

  2. B

    Tn=Ln−RnT_n=L_n-R_n

  3. C

    Tn=Ln+Rn2T_n=\frac{L_n+R_n}{2}

  4. D

    Tn=Rn−Ln2T_n=\frac{R_n-L_n}{2}

16
True or false
1 point

True or false: If F(x)=∫axf(t) dtF(x)=\int_a^x f(t)\,dt and ff is continuous, then F′(x)=f(x)F'(x)=f(x).

  1. A

    True

  2. B

    False

17
Choose one
1 point

Which expression is an antiderivative of cos⁡(3x)\cos(3x)?

  1. A

    sin⁡(3x)+C\sin(3x)+C

  2. B

    13sin⁡(3x)+C\frac{1}{3}\sin(3x)+C

  3. C

    3sin⁡(3x)+C3\sin(3x)+C

  4. D

    −13cos⁡(3x)+C-\frac{1}{3}\cos(3x)+C

18
Written response
1 point

In the notation ∫f(x) dx\int f(x)\,dx, what is the mathematical term for f(x)f(x)?

19
Written response
1 point

Evaluate exactly: ∫012x(x2+1)3 dx\int_0^1 2x(x^2+1)^3\,dx. Enter the answer as a fraction, not a decimal approximation.

20
Written response
1 point

Suppose f'(x)=6x-4 and f(1)=5. What is the value of f(2)?