Free Practice Quiz Question List

08 Antiderivatives and Indefinite Integrals Online Quiz Questions

Use this free practice quiz with 20 questions to review 08 Antiderivatives and Indefinite Integrals, test your knowledge, and prepare for your next test or exam.

20 questions

True or false

1 point

True or false: If F′(x) = f(x) on an interval, then F is an antiderivative of f on that interval.

A
True
B
False

Choose one

1 point

Which expression represents the complete family of antiderivatives of 4x³ − 6x?

A
x⁴ − 3x² + C
B
4x⁴ − 6x² + C
C
x⁴ − 6x² + C
D
12x² − 6 + C

Fill in the blank

1 point

Complete the statement: In the notation ∫ f(x) dx, f(x) is the , and dx identifies as the variable of integration.

Written response

1 point

For the initial-value problem y′ = 3x² − 4 with y(1) = 2, what is the value of the constant of integration C?

Choose one

1 point

Which expression is an antiderivative of 2x(x² + 3)⁴?

A
(x² + 3)⁵/5 + C
B
(x² + 3)⁵/10 + C
C
2(x² + 3)⁵/5 + C
D
(x² + 3)⁴/4 + C

Choose all

1 point

Select all correct indefinite-integral formulas.

A
∫eˣ dx = eˣ + C
B
∫cos x dx = sin x + C
C
∫sec²x dx = tan x + C
D
∫sin x dx = cos x + C
E
∫1/x dx = 1/x + C

Written response

1 point

An object has velocity v(t) = 6t − 4 and initial position s(0) = 3. What is the numerical value of its position s(2)?

Fill in the blank

1 point

Complete the relationships for one-dimensional motion: v(t) is the derivative of , and a(t) is the derivative of .

Choose all

1 point

Select all statements that correctly describe solving an initial-value problem of the form y′ = f(x) with y(x₀) = y₀.

A
Integrate the differential equation to obtain a general solution.
B
Use the given initial condition to determine the constant of integration.
C
Treat the initial condition as unnecessary once the differential equation is integrated.
D
Differentiate the resulting function to verify the differential equation.

Choose one

1 point

Which statement best describes the meaning of an indefinite integral?

A
A single numerical value obtained by evaluating the integrand at one point.
B
The entire family of antiderivatives F(x) + C.
C
Only the antiderivative with constant of integration C = 0.
D
The area between a curve and the x-axis over specified endpoints.

Open ended

1 point

Evaluate the indefinite integral ∫ x√(x² + 1) dx using substitution. Show the substitution, the differential adjustment, the integration, and the result after substituting back.

Choose one

1 point

Which expression is an antiderivative of 1/(1 + x²)?

A
arctan x + C
B
arcsin x + C
C
ln|1 + x²| + C
D
1/(1 + x²) + C

Choose one

1 point

What does an indefinite integral represent?

A
A single numerical value obtained by evaluating a function at two endpoints
B
The entire family of antiderivatives F(x) + C
C
Only the derivative of the integrand
D
A function that must equal zero at x = 0

True or false

1 point

True or false: The power rule can be applied directly to ∫ 1/x dx by using n = −1.

A
True
B
False

Choose one

1 point

Which expression is the general antiderivative of 2x² − 6x + 4?

A
2x³ − 3x² + 4x + C
B
6x³ − 6x² + 4 + C
C
(2/3)x³ − 3x² + 4x + C
D
(1/2)x⁴ − 3x² + 4x + C

Written response

1 point

For the initial-value problem dy/dx = 4x³ − 2x with y(1) = 5, what is the numerical value of the constant of integration C?

Choose one

1 point

If acceleration a(t) is known and both velocity and position are required, which procedure is correct?

A
Integrate a(t) once to obtain v(t), then integrate v(t) to obtain s(t).
B
Integrate a(t) once to obtain s(t) directly.
C
Differentiate a(t) once to obtain v(t).
D
Integrate s(t) to obtain v(t), then differentiate v(t) to obtain a(t).

Written response

1 point

What integration method reverses the chain rule by setting an inner expression equal to u?

Choose one

1 point

Which is the correct indefinite integral of 1/x with respect to x?

A
x⁻¹ + C
B
−1/x² + C
C
|x| + C
D
ln|x| + C

True or false

1 point

True or false: If $F'(x)=f(x)$ and $G'(x)=g(x)$ on an interval, then $2F(x)+3G(x)$ is an antiderivative of $2f(x)+3g(x)$ on that interval.

A
True
B
False