Question 1

Which of the following best describes the most general antiderivative of a function $$f(x)$$?

Accepted Answer

$$F(x) + C$$, where $$F'(x) = f(x)$$ and $$C is $$any real constant

Question 2

Evaluate the indefinite integral: $$\int 7x^3$ dx.$$

Accepted Answer

$$\dfrac{7}{4}x^4$ + C$$

Question 3

Which of the following is NOT a correct application of the power rule for integration?

Accepted Answer

$$\int x$$^{-1}$$ dx = \dfrac{x^0$}{0} + C$$

Question 4

A medication's concentration in the bloodstream is modeled by $$f'(t) = -2t mg/L $$per hour. If the initial concentration is 10 mg/L, what is the most general formula for the concentration $$f(t)$$?

Accepted Answer

$$-t^2 + C$

Question 5

Which property of integration allows you to write $$\int [3x^2$ - 5x] dx as$$ $$3\int x^2$ dx - 5\int x dx$$?

Accepted Answer

Linearity of integration

Question 6

Rewrite and evaluate $$\int \sqrt[3]{x^2$} dx.$$

Accepted Answer

$$\dfrac{3}{5}x$$^{5/3}$$ + C$$

Question 7

A car's velocity is given by $$v(t) = 4t^3$ - 2t. $$What is the most general formula for its position $$s(t)$$?

Accepted Answer

$$s(t) = t^4$ - t^2$ + C$$

Question 8

Which of the following is the correct antiderivative of $$f(x) = \dfrac{1}{x^4$}$$?

Accepted Answer

$$-\dfrac{1}{3}x$$^{-3}$$ + C$$

Question 9

Evaluate $$\int (2x$$^{-2}$$ + 5x$$^{1/2}$$) dx.$$

Accepted Answer

$$-2x^{-1}$$ + \dfrac{10}{3}$$x^{3/2} + C$

Question 10

Which step is necessary before integrating $$\int \dfrac{5}{x^3$} dx$$ using the power rule?

Accepted Answer

Rewrite $$\dfrac{5}{x^3$} as$$ $$5x^{-3}$$

Question 11

A population grows at a rate $$P'(t) = 12t^2$ + 7. $$What is the most general formula for the population $$P(t)$$?

Accepted Answer

$$4t^3 + 7t + C$

Question 12

Which of the following is a correct verification step after finding an antiderivative $$F(x)$$ for $$f(x)$$?

Accepted Answer

Differentiate $$F(x)$$ and check if $$F'(x) = f(x)$$

Question 13

Evaluate $$\int (6x^2$ - 4x + 9) dx.$$

Accepted Answer

$$2x^3$$ - $$2x^2 + 9x + C$

Question 14

Which of the following integrals requires a different technique than the power rule?

Accepted Answer

$$\int \dfrac{1}{x} dx$$

Question 15

A chemical reaction's rate is $$r(t) = 0.5t$$^{-1/2}$$. Find the most general formula for the amount of product formed, A(t$$)$$.

Accepted Answer

$$A(t) = t$$^{1/2}$$ + C$$

Question 16

Evaluate $$\int 10 dx.$$

Accepted Answer

$$10x + C$$

Question 17

Which of the following is a common mistake when integrating $$\int (x^2$ + 3x) dx$$?

Accepted Answer

Forgetting to add the constant of integration $$C$$

Question 18

Evaluate $$\int \dfrac{3}{10}x^5$ dx.$$

Accepted Answer

$$\dfrac{1}{20}x^6$ + C$$

Question 19

Which of the following best explains why $$\int (x^2$ + 2x) dx$$ can be computed as $$\int x^2$ dx + \int 2x dx$$?

Accepted Answer

Because integration is linear and distributes over addition

Antiderivatives and Indefinite Integration: Concepts and Techniques

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