Question 1

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

Accepted Answer

A function $F(x)$ such that $F'(x) = f(x)$ for all $x$ in the interval.

Question 2

What is the most general form of the indefinite integral of $f(x)$?

Accepted Answer

$\int f(x) dx = F(x) + C$, where $C$ is an arbitrary constant.

Question 3

Evaluate $\int 5x^4 dx$.

Accepted Answer

$x^5 + C$

Question 4

Which rule allows you to integrate a sum of functions?

Accepted Answer

$\int (af(x) + bg(x)) dx = a \int f(x) dx + b \int g(x) dx$

Question 5

Find $\int \frac{1}{x} dx$.

Accepted Answer

$\ln|x| + C$

Question 6

Evaluate $\int e^{ax} dx$ where $a 
eq 0$.

Accepted Answer

$\frac{e^{ax}}{a} + C$

Question 7

A company’s marginal revenue is $R'(q) = 4000 - 8q^{-0.8}$ (dollars per unit). What is the revenue function $R(q)$, assuming $R(0) = 0$?

Accepted Answer

$R(q) = 4000q + 40q^{0.2}$

Question 8

An object is thrown downward from a tower with acceleration $a(t) = -32$ ft/s$^2$ and initial velocity $v(0) = -20$ ft/s. What is the velocity function $v(t)$?

Accepted Answer

$v(t) = -32t - 20$

Question 9

Given $v(t) = -32t - 20$ and initial height $s(0) = 1100$ ft, what is the position function $s(t)$?

Accepted Answer

$s(t) = -16t^2 - 20t + 1100$

Question 10

Which substitution would be most appropriate for evaluating $\int 6x(3x^2 + 4)^7 dx$?

Accepted Answer

Let $u = 3x^2 + 4$

Question 11

Evaluate $\int x^2 \sqrt{x^3 + 1} dx$ using substitution.

Accepted Answer

$\frac{2}{9}(x^3 + 1)^{3/2} + C$

Question 12

Find $\int \frac{x+3}{x^2 + 6x} dx$ using substitution.

Accepted Answer

$\frac{1}{2} \ln|x^2 + 6x| + C$

Question 13

Evaluate $\int \frac{2x - 3}{x^2 - 3x} dx$ using substitution.

Accepted Answer

$\ln|x^2 - 3x| + C$

Question 14

What is the result of $\int 3x e^{x^2} dx$?

Accepted Answer

$e^{x^2} + C$

Question 15

Which substitution is appropriate for $\int x \sqrt{1 - x} dx$?

Accepted Answer

Let $u = 1 - x$

Question 16

After substituting $u = 1 - x$ in $\int x \sqrt{1 - x} dx$, what does $x$ become in terms of $u$?

Accepted Answer

$x = 1 - u$

Question 17

Which of the following is a correct application of the substitution rule for $\int f(g(x))g'(x) dx$?

Accepted Answer

Let $u = g(x)$, then $du = g'(x) dx$, so $\int f(g(x))g'(x) dx = \int f(u) du$

Question 18

Which of the following is NOT a typical choice for $u$ in substitution problems?

Accepted Answer

A constant term in the integrand

Antiderivatives and Substitution: Business Calculus Study Notes

Study Guide - Practice Questions

Study Guide - Flashcards