Question 1

Which of the following best describes the relationship between differentiation and integration?

Accepted Answer

Integration is the inverse operation of differentiation.

Question 2

Given $$f(x) = 3x^2$, what is the general form of its antiderivative?

Accepted Answer

$$F(x) = x^3$ + C$$

Question 3

Which of the following is the correct indefinite integral for $$\int x^4$ dx$$?

Accepted Answer

$$\frac{1}{5}x^5$ + C$$

Question 4

A ball is thrown upward with an initial velocity of 19.6 m/s from a height of 24.5 m. Assuming acceleration due to gravity is $$a(t) = -9.8 m/s$^2$$, what is the position function $$s(t)$$?

Accepted Answer

$$s(t) = -4.9t^2$ + 19.6t + 24.5$$

Question 5

Find the indefinite integral $$\int \frac{1}{x^3$} dx.$$

Accepted Answer

$$-\frac{1}{2x^2$} + C$$

Question 6

If $$F'(x) = 2x$$ and $$F(1) = 5$$, what is the particular solution for $$F(x)$$?

Accepted Answer

$$F(x) = x^2$ + 4$$

Question 7

Which of the following is NOT a valid basic integration rule?

Accepted Answer

$$\int f(x)g(x) dx = (\int f(x) dx)(\int g(x) dx)$$

Question 8

A car accelerates from rest with constant acceleration $$a = 2 m/s$^2. $$What is the velocity function $$v(t) if$$ $$v(0) = 0$$?

Accepted Answer

$$v(t) = 2t$$

Question 9

Given $$\int \sin x dx$$, which of the following is the correct antiderivative?

Accepted Answer

$$-\cos x + C$$

Question 10

Suppose $$f''(x) = 2$$, $$f'(2) = 5$$, and $$f(2) = 10. $$What is the particular solution $$f(x)$$?

Accepted Answer

$$f(x) = x^2$ + x + 6$$

Question 11

Which of the following statements is TRUE about antiderivatives?

Accepted Answer

Any two antiderivatives of the same function differ by a constant.

Question 12

Find the indefinite integral $$\int (3x^4$ - 5x^2$ + x) dx.$$

Accepted Answer

$$\frac{3}{5}x^5$ - \frac{5}{3}x^3$ + \frac{1}{2}x^2$ + C$$

Question 13

A population of bacteria grows at a rate $$\frac{dP}{dt} = k\sqrt{t}$$, with $$P(0) = 500. $$What is the general solution for $$P(t)$$?

Accepted Answer

$$P(t) = \frac{2k}{3}t$$^{3/2}$$ + 500$$

Question 14

Which of the following is the correct indefinite integral for $$\int \sec^2$ x dx$$?

Accepted Answer

$$	an x + C$$

Question 15

If $$\int f(x) dx = F(x) + C$$, what does $$C$$ represent?

Accepted Answer

The constant of integration

Question 16

A stone is dropped from rest from a cliff on the moon, where $$a(t) = -1.6 m/s$^2. If it $$hits the surface 20 seconds later, how far did it fall?

Accepted Answer

$$320$$ meters

Question 17

Which of the following is the correct antiderivative for $$\int \cos x dx$$?

Accepted Answer

$$\sin x + C$$

Question 18

Given $$\int (x + 1)(3x - 2) dx$$, what is the correct first step to solve this integral?

Accepted Answer

Expand the integrand to $$3x^2 + x - 2$

Antiderivatives and Indefinite Integration: Foundations of Integral Calculus

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