Q: Which rule allows you to find the antiderivative of a sum of functions?

Sum or Difference Rule: $\\int [f(x) + g(x)] dx = \\int f(x) dx + \\int g(x) dx$

Q: Find the indefinite integral: $\\int (2x^3 - 5x + 6) dx$

$\\frac{2}{4}x^4 - \\frac{5}{2}x^2 + 6x + C$

Q: If $f(x) = 5x^2 - 3x + 2$, what is $\\int f(x) dx$?

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

Q: If $f(x) = 7x^2 - 4x + 9$, what is $\\int f(x) dx$?

$\\frac{7}{3}x^3 - 2x^2 + 9x + C$

Question 1

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

Accepted Answer

A function $F(x)$ such that $F'(x) = f(x)$.

Question 2

What is the general form of the indefinite integral of $f(x) = 3x^2$?

Accepted Answer

$rac{3}{3}x^{3} + C$

Question 3

Which rule allows you to find the antiderivative of a sum of functions?

Accepted Answer

Sum or Difference Rule: $\int [f(x) + g(x)] dx = \int f(x) dx + \int g(x) dx$

Question 4

Find the indefinite integral: $\int (2x^3 - 5x + 6) dx$

Accepted Answer

$\frac{2}{4}x^4 - \frac{5}{2}x^2 + 6x + C$

Question 5

Which of the following is the correct antiderivative of $f(x) = e^{2x}$?

Accepted Answer

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

Question 6

A company’s marginal cost function is $C'(x) = 2x + 5$. If the fixed cost is $10$, what is the total cost function $C(x)$?

Accepted Answer

$x^2 + 5x + 10$

Question 7

Find $F(x)$ if $F'(x) = 4x^3 - 2x + 7$ and $F(0) = 5$.

Accepted Answer

$x^4 - x^2 + 7x + 5$

Question 8

Which of the following is the correct indefinite integral for $\int \frac{1}{x} dx$?

Accepted Answer

$\ln|x| + C$

Question 9

If $f(x) = 5x^2 - 3x + 2$, what is $\int f(x) dx$?

Accepted Answer

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

Question 10

A business finds its marginal profit function is $P'(x) = \sqrt{x}$, where $x$ is the number of units sold. If the profit is $1000$ when $x = 0$, what is the profit function $P(x)$?

Accepted Answer

$\frac{2}{3}x^{3/2} + 1000$

Question 11

Which of the following is the correct antiderivative for $f(x) = a^x$, where $a > 0$ and $a \neq 1$?

Accepted Answer

$\frac{a^x}{\ln a} + C$

Question 12

Find the indefinite integral: $\int (x^{-2} + 4x^3) dx$

Accepted Answer

$-x^{-1} + x^4 + C$

Question 13

Suppose $C'(x) = 3x^2$ and $C(2) = 10$. What is $C(x)$?

Question 14

Which of the following is the correct indefinite integral for $\int \frac{1}{x} dx$?

Accepted Answer

$\ln|x| + C$

Question 15

If $f(x) = 2e^{3x}$, what is $\int f(x) dx$?

Accepted Answer

$\frac{2}{3}e^{3x} + C$

Question 16

A company’s marginal revenue function is $R'(x) = 4x - 7$. If $R(0) = 20$, what is the revenue function $R(x)$?

Accepted Answer

$2x^2 - 7x + 20$

Question 17

Find the indefinite integral: $\int (5x^4 - 2x + 1) dx$

Accepted Answer

$x^5 - x^2 + x + C$

Question 18

Which of the following is the correct antiderivative for $f(x) = \frac{1}{x}$?

Accepted Answer

$\ln|x| + C$

Question 19

If $f(x) = 7x^2 - 4x + 9$, what is $\int f(x) dx$?

Accepted Answer

$\frac{7}{3}x^3 - 2x^2 + 9x + C$

Section 7.1: Antiderivatives – Business Calculus Study Notes

Study Guide - Practice Questions

Study Guide - Flashcards