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Calculus for Business, Economics, Life Sciences, and Social Sciences 13th

Raymond A. Barnett, Michael R. Ziegler, Karl E. Byleen

Chapter 5

Integration

Educators


Problem 1

Write each function as a sum of terms of the form $a x^{n}$, where a is a constant. (If necessary, review Section A.6).
$$
f(x)=\frac{5}{x^{4}}
$$

Priyanka S.
Numerade Educator

Problem 2

Write each function as a sum of terms of the form $a x^{n}$, where a is a constant. (If necessary, review Section A.6).
$$
f(x)=-\frac{6}{x^{9}}
$$

Priyanka S.
Numerade Educator

Problem 3

Write each function as a sum of terms of the form $a x^{n}$, where a is a constant. (If necessary, review Section A.6).
$$
f(x)=\frac{3 x-2}{x^{5}}
$$

Priyanka S.
Numerade Educator

Problem 4

Write each function as a sum of terms of the form $a x^{n}$, where a is a constant. (If necessary, review Section A.6).
$$
f(x)=\frac{x^{2}+5 x-1}{x^{3}}
$$

Priyanka S.
Numerade Educator

Problem 5

Write each function as a sum of terms of the form $a x^{n}$, where a is a constant. (If necessary, review Section A.6).
$$
f(x)=\sqrt{x}+\frac{5}{\sqrt{x}}
$$

Priyanka S.
Numerade Educator

Problem 6

Write each function as a sum of terms of the form $a x^{n}$, where a is a constant. (If necessary, review Section A.6).
$$
f(x)=\sqrt[3]{x}-\frac{4}{\sqrt[3]{x}}
$$

Priyanka S.
Numerade Educator

Problem 7

Write each function as a sum of terms of the form $a x^{n}$, where a is a constant. (If necessary, review Section A.6).
$$
f(x)=\sqrt[3]{x}\left(4+x-3 x^{2}\right)
$$

Priyanka S.
Numerade Educator

Problem 8

Write each function as a sum of terms of the form $a x^{n}$, where a is a constant. (If necessary, review Section A.6).
$$
f(x)=\sqrt{x}\left(1-5 x+x^{3}\right)
$$

Priyanka S.
Numerade Educator

Problem 9

Find each indefinite integral. Check by differentiating.
$$
\int 7 d x
$$

Priyanka S.
Numerade Educator

Problem 10

Find each indefinite integral. Check by differentiating.
$$
\int 10 d x
$$

Priyanka S.
Numerade Educator

Problem 11

Find each indefinite integral. Check by differentiating.
$$
\int 8 x d x
$$

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Numerade Educator

Problem 12

Find each indefinite integral. Check by differentiating.
$$
\int 14 x d x
$$

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Numerade Educator

Problem 13

Find each indefinite integral. Check by differentiating.
$$
\int 9 x^{2} d x
$$

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Numerade Educator

Problem 14

Find each indefinite integral. Check by differentiating.
$$
\int 15 x^{2} d x
$$

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Numerade Educator

Problem 15

Find each indefinite integral. Check by differentiating.
$$
\int x^{5} d x
$$

Priyanka S.
Numerade Educator

Problem 16

Find each indefinite integral. Check by differentiating.
$$
\int x^{8} d x
$$

Priyanka S.
Numerade Educator

Problem 17

Find each indefinite integral. Check by differentiating.
$$
\int x^{-3} d x
$$

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Numerade Educator

Problem 18

Find each indefinite integral. Check by differentiating.
$$
\int x^{-4} d x
$$

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Numerade Educator

Problem 19

Find each indefinite integral. Check by differentiating.
$$
\int 10 x^{3 / 2} d x
$$

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Numerade Educator

Problem 20

Find each indefinite integral. Check by differentiating.
$$
\int 8 x^{1 / 3} d x
$$

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Numerade Educator

Problem 21

Find each indefinite integral. Check by differentiating.
$$
\int \frac{3}{z} d z
$$

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Problem 22

Find each indefinite integral. Check by differentiating.
$$
\int \frac{7}{z} d z
$$

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Numerade Educator

Problem 23

Find each indefinite integral. Check by differentiating.
$$
\int 16 e^{u} d u
$$

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Problem 24

Find each indefinite integral. Check by differentiating.
$$
\int 5 e^{u} d u
$$

Priyanka S.
Numerade Educator

Problem 25

Is $F(x)=(x+1)(x+2)$ an antiderivative of $f(x)=2 x+3 ?$ Explain.

Priyanka S.
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Problem 26

Is $F(x)=(2 x+5)(x-6)$ an antiderivative of $f(x)=4 x-7 ?$ Explain.

Priyanka S.
Numerade Educator

Problem 27

Is $F(x)=1+x \ln x$ an antiderivative of $f(x)=1+\ln x ?$
Explain.

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Numerade Educator

Problem 28

Is $F(x)=x$ in $x-x+e$ an antiderivative of $f(x)=\ln x ?$ Explain.

Priyanka S.
Numerade Educator

Problem 29

Is $F(x)=\frac{(2 x+1)^{3}}{3}$ an antiderivative of $f(x)=(2 x+1)^{2} ?$ Explain.

Priyanka S.
Numerade Educator

Problem 30

Is $F(x)=\frac{(3 x-2)^{4}}{4}$ an antiderivative of $f(x)=(3 x-2)^{3} ?$ Explain.

Priyanka S.
Numerade Educator

Problem 31

Is $F(x)=e^{x^{3} / 3}$ an antiderivative of $f(x)=e^{x^{2}} ?$ Explain.

Priyanka S.
Numerade Educator

Problem 32

Is $F(x)=\left(e^{x}-10\right)\left(e^{x}+10\right)$ an antiderivative of $f(x)=2 e^{2 x}$ ? Explain.

Priyanka S.
Numerade Educator

Problem 33

Discuss the validity of each statement. If the statement is always true, explain why. If not, give a counterexample.
The constant function $f(x)=\pi$ is an antiderivative of the constant function $k(x)=0$.

Priyanka S.
Numerade Educator

Problem 34

Discuss the validity of each statement. If the statement is always true, explain why. If not, give a counterexample.
The constant function $k(x)=0$ is an antiderivative of the constant function $f(x)=\pi$.

Priyanka S.
Numerade Educator

Problem 35

Discuss the validity of each statement. If the statement is always true, explain why. If not, give a counterexample.
If $n$ is an integer, then $x^{n+1} /(n+1)$ is an antiderivative of $x^{n}$.

Priyanka S.
Numerade Educator

Problem 36

Discuss the validity of each statement. If the statement is always true, explain why. If not, give a counterexample.
The constant function $k(x)=0$ is an antiderivative of itself.

Priyanka S.
Numerade Educator

Problem 37

Discuss the validity of each statement. If the statement is always true, explain why. If not, give a counterexample.
The function $h(x)=5 e^{x}$ is an antiderivative of itself.

Priyanka S.
Numerade Educator

Problem 38

Discuss the validity of each statement. If the statement is always true, explain why. If not, give a counterexample.
The constant function $g(x)=5 e^{\pi}$ is an antiderivative of itself.

Priyanka S.
Numerade Educator

Problem 39

Could the three graphs in each figure be antiderivatives of the same function? Explain.

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Numerade Educator

Problem 40

Could the three graphs in each figure be antiderivatives of the same function? Explain.

Priyanka S.
Numerade Educator

Problem 41

Could the three graphs in each figure be antiderivatives of the same function? Explain.

Priyanka S.
Numerade Educator

Problem 42

Could the three graphs in each figure be antiderivatives of the same function? Explain.

Priyanka S.
Numerade Educator

Problem 43

Find each indefinite integral. (Check by differentiation.)
$$
\int 5 x(1-x) d x
$$

Priyanka S.
Numerade Educator

Problem 44

Find each indefinite integral. (Check by differentiation.)
$$
\int x^{2}\left(1+x^{3}\right) d x
$$

Priyanka S.
Numerade Educator

Problem 45

Find each indefinite integral. (Check by differentiation.)
$$
\int \frac{d u}{\sqrt{u}}
$$

Priyanka S.
Numerade Educator

Problem 46

Find each indefinite integral. (Check by differentiation.)
$$
\int \frac{d t}{\sqrt[3]{t}}
$$

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Numerade Educator

Problem 47

Find each indefinite integral. (Check by differentiation.)
$$
\int \frac{d x}{4 x^{3}}
$$

Priyanka S.
Numerade Educator

Problem 48

Find each indefinite integral. (Check by differentiation.)
$$
\int \frac{6 d m}{m^{2}}
$$

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Numerade Educator

Problem 49

Find each indefinite integral. (Check by differentiation.)
$$
\int \frac{4+u}{u} d u
$$

Priyanka S.
Numerade Educator

Problem 50

Find each indefinite integral. (Check by differentiation.)
$$
\int \frac{1-y^{2}}{3 y} d y
$$

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Numerade Educator

Problem 51

Find each indefinite integral. (Check by differentiation.)
$$
\int\left(5 e^{z}+4\right) d z
$$

Priyanka S.
Numerade Educator

Problem 52

Find each indefinite integral. (Check by differentiation.)
$$
\int \frac{e^{t}-t}{2} d t
$$

Priyanka S.
Numerade Educator

Problem 53

Find each indefinite integral. (Check by differentiation.)
$$
\int\left(3 x^{2}-\frac{2}{x^{2}}\right) d x
$$

Priyanka S.
Numerade Educator

Problem 54

Find each indefinite integral. (Check by differentiation.)
$$
\int\left(4 x^{3}+\frac{2}{x^{3}}\right) d x
$$

Priyanka S.
Numerade Educator

Problem 55

Find the particular antiderivative of each derivative that satisfies the given condition.
$$
C^{\prime}(x)=6 x^{2}-4 x ; C(0)=3,000
$$

Priyanka S.
Numerade Educator

Problem 56

Find the particular antiderivative of each derivative that satisfies the given condition.
$$
R^{\prime}(x)=600-0.6 x ; R(0)=0
$$

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Numerade Educator

Problem 57

Find the particular antiderivative of each derivative that satisfies the given condition.
$$
\frac{d x}{d t}=\frac{20}{\sqrt{t}} ; x(1)=40
$$

Priyanka S.
Numerade Educator

Problem 58

Find the particular antiderivative of each derivative that satisfies the given condition.
$$
\frac{d R}{d t}=\frac{100}{t^{2}} ; R(1)=400
$$

Priyanka S.
Numerade Educator

Problem 59

Find the particular antiderivative of each derivative that satisfies the given condition.
$$
\frac{d y}{d x}=2 x^{-2}+3 x^{-1}-1 ; y(1)=0
$$

Priyanka S.
Numerade Educator

Problem 60

Find the particular antiderivative of each derivative that satisfies the given condition.
$$
\frac{d y}{d x}=3 x^{-1}+x^{-2} ; y(1)=1
$$

Priyanka S.
Numerade Educator

Problem 61

Find the particular antiderivative of each derivative that satisfies the given condition.
$$
\frac{d x}{d t}=4 e^{t}-2 ; x(0)=1
$$

Priyanka S.
Numerade Educator

Problem 62

Find the particular antiderivative of each derivative that satisfies the given condition.
$$
\frac{d y}{d t}=5 e^{t}-4 ; y(0)=-1
$$

Priyanka S.
Numerade Educator

Problem 63

Find the equation of the curve that passes through (2,3) if its slope is given by $\frac{d y}{d x}=4 x-3$ for each $x$.

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Problem 64

Find the equation of the curve that passes through (1,3) if its slope is given by $\frac{d y}{d x}=12 x^{2}-12 x$ for each $x$.

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Problem 65

Find each indefinite integral.
$\int \frac{2 x^{4}-x}{x^{3}} d x$

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Numerade Educator

Problem 66

Find each indefinite integral.
$\int \frac{x^{-1}-x^{4}}{x^{2}} d x$

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Numerade Educator

Problem 67

Find each indefinite integral.
$\int \frac{x^{5}-2 x}{x^{4}} d x$

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Problem 68

Find each indefinite integral.
$\int \frac{1-3 x^{4}}{x^{2}} d x$

Priyanka S.
Numerade Educator

Problem 69

Find each indefinite integral.
$\int \frac{x^{2} e^{x}-2 x}{x^{2}} d x$

Priyanka S.
Numerade Educator

Problem 70

Find each indefinite integral.
$\int \frac{1-x e^{x}}{x} d x$

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Problem 71

Find the derivative or indefinite integral as indicated.
$\frac{d}{d x}\left(\int x^{3} d x\right)$

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Problem 72

Find the derivative or indefinite integral as indicated.
$\frac{d}{d t}\left(\int \frac{\ln t}{t} d t\right)$

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Problem 73

Find the derivative or indefinite integral as indicated.
$\int \frac{d}{d x}\left(x^{4}+3 x^{2}+1\right) d x$

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Numerade Educator

Problem 74

Find the derivative or indefinite integral as indicated.
$\int \frac{d}{d u}\left(e^{u^{2}}\right) d u$

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Problem 75

Use differentiation to justify the formula $\int x^{n} d x=\frac{x^{n+1}}{n+1}+C$ provided that $n \neq-1$.

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Problem 76

Use differentiation to justify the formula $\int e^{x} d x=e^{x}+C$

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Problem 77

Assuming that $x>0$, use differentiation to justify the formula $\int \frac{1}{x} d x=\ln |x|+C$

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Problem 78

Assuming that $x<0$, use differentiation to justify the formula $\int \frac{1}{x} d x=\ln |x|+C$
[Hint: Use the chain rule after noting that $\ln |x|=\ln (-x)$ for $x<0 .]$

Priyanka S.
Numerade Educator

Problem 79

Show that the indefinite integral of the sum of two functions is the sum of the indefinite integrals.
[Hint: Assume that $\int f(x) d x=F(x)+C_{1}$ and $\int g(x) d x=G(x)+C_{2}$. Using differentiation, show that $F(x)+C_{1}+G(x)+C_{2}$ is the indefinite integral of the function $s(x)=f(x)+g(x)$.]

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Problem 80

Show that the indefinite integral of the difference of two functions is the difference of the indefinite integrals.

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Numerade Educator

Problem 81

The marginal average cost of producing $x$ sports watches is given by $\bar{C}^{\prime}(x)=-\frac{1,000}{x^{2}} \quad \bar{C}(100)=25$ where $\bar{C}(x)$ is the average cost in dollars. Find the average cost function and the cost function. What are the fixed costs?

Priyanka S.
Numerade Educator

Problem 82

In 2012, U.S. consumption of renewable energy was 8.45 quadrillion Btu (or $8.45 \times 10^{15} \mathrm{Btu}$ ). Since the 1960 s, consumption has been growing at a rate (in quadrillion Btu per year) given by $f^{\prime}(t)=0.004 t+0.062$ where $t$ is years after $1960 .$ Find $f(t)$ and estimate U.S. consumption of renewable energy in $2024 .$

Check back soon!

Problem 83

The graph of the marginal cost function from the production of $x$ thousand bottles of sunscreen per month [where cost $C(x)$ is in thousands of dollars per month] is given in the figure.
(A) Using the graph shown, describe the shape of the graph of the cost function $C(x)$ as $x$ increases from 0 to 8,000 bottles per month.
(B) Given the equation of the marginal cost function, $C^{\prime}(x)=3 x^{2}-24 x+53$ find the cost function if monthly fixed costs at 0 output are $\$ 30,000 .$ What is the cost of manufacturing 4,000 bottles per month? 8,000 bottles per month?
(C) Graph the cost function for $0 \leq x \leq 8$. [Check the shape of the graph relative to the analysis in part (A).]
(D) Why do you think that the graph of the cost function is steeper at both ends than in the middle?

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Problem 84

The graph of the marginal revenue function from the sale of $x$ sports watches is given in the figure.
(A) Using the graph shown, describe the shape of the graph of the revenue function $R(x)$ as $x$ increases from 0 to 1,000
(B) Find the equation of the marginal revenue function (the linear function shown in the figure).
(C) Find the equation of the revenue function that satisfies $R(0)=0$. Graph the revenue function over the interva $[0,1,000] .[$ Check the shape of the graph relative to the analysis in part (A).
(D) Find the price-demand equation and determine the price when the demand is 700 units.

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Problem 85

Monthly sales of an SUV model are expected to increase at the rate of $S^{\prime}(t)=-24 t^{1 / 3}$
SUVs per month, where $t$ is time in months and $S(t)$ is the number of SUVs sold each month. The company plans to stop manufacturing this model when monthly sales reach $300 \mathrm{SUVs} .$ If monthly sales now $(t=0)$ are $1,200 \mathrm{SUVs}$ find $S(t) .$ How long will the company continue to manufacture this model?

Priyanka S.
Numerade Educator

Problem 86

The rate of change of the monthly sales of a newly released football game is given by $S^{\prime}(t)=500 t^{1 / 4} \quad S(0)=0$ where $t$ is the number of months since the game was released and $S(t)$ is the number of games sold each month. Find $S(t)$. When will monthly sales reach 20,000 games?

Priyanka S.
Numerade Educator

Problem 87

Repeat Problem 85 if $S^{\prime}(t)=-24 t^{1 / 3}-70$ and all other information remains the same. Use a graphing calculator to approximate the solution of the equation $S(t)=300$ to two decimal places.

Priyanka S.
Numerade Educator

Problem 88

Repeat Problem 86 if $S^{\prime}(t)=500 t^{1 / 4}+300$ and all other information remains the same. Use a graphing calculator to approximate the solution of the equation $S(t)=20,000$ to two decimal places.

Priyanka S.
Numerade Educator

Problem 89

A defense contractor is starting production on a new missile control system. On the basis of data collected during the assembly of the first 16 control systems, the production manager obtained the following function describing the rate of labor use: $L^{\prime}(x)=2,400 x^{-1 / 2}$
For example, after assembly of 16 units, the rate of assembly is 600 labor-hours per unit, and after assembly of 25 units, the rate of assembly is 480 labor-hours per unit. The more units assembled, the more efficient the process. If 19,200 labor-hours are required to assemble of the first 16 units, how many labor-hours $L(x)$ will be required to assemble the first $x$ units? The first 25 units?

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Numerade Educator

Problem 90

If the rate of labor use in Problem 89 is $L^{\prime}(x)=2,000 x^{-1 / 3}$ and if the first 8 control units require 12,000 labor-hours, how many labor-hours, $L(x),$ will be required for the first $x$ control units? The first 27 control units?

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Numerade Educator

Problem 91

For an average person, the rate of change of weight $W$ (in pounds) with respect to height $h$ (in inches) is given approximately by $\frac{d W}{d h}=0.0015 h^{2}$ Find $W(h)$ if $W(60)=108$ pounds. Find the weight of an average person who is 5 feet, 10 inches, tall.

Priyanka S.
Numerade Educator

Problem 92

The area $A$ of a healing wound changes at a rate given approximately by $\frac{d A}{d t}=-4 t^{-3} \quad 1 \leq t \leq 10$ where $t$ is time in days and $A(1)=2$ square centimeters. What will the area of the wound be in 10 days?

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Numerade Educator

Problem 93

The rate of growth of the population $N(t)$ of a new city $t$ years after its incorporation is estimated to be $\frac{d N}{d t}=400+600 \sqrt{t} \quad 0 \leq t \leq 9$
If the population was 5,000 at the time of incorporation, find the population 9 years later.

Priyanka S.
Numerade Educator

Problem 94

A college language class was chosen for an experiment in learning. Using a list of 50 words, the experiment involved measuring the rate of vocabulary memorization at different times during a continuous 5 -hour study session. It was found that the average rate of learning for the entire class was inversely proportional to the time spent studying and was given approximately by $V^{\prime}(t)=\frac{15}{t} \quad 1 \leq t \leq 5$
If the average number of words memorized after 1 hour of study was 15 words, what was the average number of words memorized after $t$ hours of study for $1 \leq t \leq 5 ?$ After 4 hours of study? Round answer to the nearest whole number.

Priyanka S.
Numerade Educator