🎉 Announcing Numerade's $26M Series A, led by IDG Capital!Read how Numerade will revolutionize STEM Learning # Calculus for Business, Economics, Life Sciences, and Social Sciences 13th ## Raymond A. Barnett, Michael R. Ziegler, Karl E. Byleen ## Chapter 5 ## Integration ## Educators ### Problem 1 Write each function as a sum of terms of the form$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). $$f(x)=\sqrt{x}-\frac{4}{\sqrt{x}}$$ 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). $$f(x)=\sqrt{x}\left(4+x-3 x^{2}\right)$$ 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. $$\int 8 x d x$$ Priyanka S. Numerade Educator ### Problem 12 Find each indefinite integral. Check by differentiating. $$\int 14 x d x$$ Priyanka S. Numerade Educator ### Problem 13 Find each indefinite integral. Check by differentiating. $$\int 9 x^{2} d x$$ Priyanka S. Numerade Educator ### Problem 14 Find each indefinite integral. Check by differentiating. $$\int 15 x^{2} d x$$ Priyanka S. 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. $$\int x^{-3} d x$$ Priyanka S. Numerade Educator ### Problem 18 Find each indefinite integral. Check by differentiating. $$\int x^{-4} d x$$ Priyanka S. Numerade Educator ### Problem 19 Find each indefinite integral. Check by differentiating. $$\int 10 x^{3 / 2} d x$$ Priyanka S. Numerade Educator ### Problem 20 Find each indefinite integral. Check by differentiating. $$\int 8 x^{1 / 3} d x$$ Priyanka S. Numerade Educator ### Problem 21 Find each indefinite integral. Check by differentiating. $$\int \frac{3}{z} d z$$ Priyanka S. Numerade Educator ### Problem 22 Find each indefinite integral. Check by differentiating. $$\int \frac{7}{z} d z$$ Priyanka S. Numerade Educator ### Problem 23 Find each indefinite integral. Check by differentiating. $$\int 16 e^{u} d u$$ Priyanka S. Numerade Educator ### 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. Numerade Educator ### 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. Priyanka S. 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. Priyanka S. 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.) $$\int \frac{d t}{\sqrt{t}}$$ Priyanka S. 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.) $$\int \frac{6 d m}{m^{2}}$$ Priyanka S. 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.) $$\int \frac{1-y^{2}}{3 y} d y$$ Priyanka S. 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. $$R^{\prime}(x)=600-0.6 x ; R(0)=0$$ Priyanka S. 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$x$. Priyanka S. Numerade Educator ### 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$x$. Priyanka S. Numerade Educator ### Problem 65 Find each indefinite integral.$\int \frac{2 x^{4}-x}{x^{3}} d x$ Priyanka S. Numerade Educator ### Problem 66 Find each indefinite integral.$\int \frac{x^{-1}-x^{4}}{x^{2}} d x$ Priyanka S. Numerade Educator ### Problem 67 Find each indefinite integral.$\int \frac{x^{5}-2 x}{x^{4}} d x$ Priyanka S. Numerade Educator ### 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.$\int \frac{1-x e^{x}}{x} d x$ Priyanka S. Numerade Educator ### Problem 71 Find the derivative or indefinite integral as indicated.$\frac{d}{d x}\left(\int x^{3} d x\right)$ Priyanka S. Numerade Educator ### Problem 72 Find the derivative or indefinite integral as indicated.$\frac{d}{d t}\left(\int \frac{\ln t}{t} d t\right)$ Priyanka S. Numerade Educator ### Problem 73 Find the derivative or indefinite integral as indicated.$\int \frac{d}{d x}\left(x^{4}+3 x^{2}+1\right) d x$ Priyanka S. Numerade Educator ### Problem 74 Find the derivative or indefinite integral as indicated.$\int \frac{d}{d u}\left(e^{u^{2}}\right) d u$ Priyanka S. Numerade Educator ### Problem 75 Use differentiation to justify the formula$\int x^{n} d x=\frac{x^{n+1}}{n+1}+C$provided that$n \neq-1$. Priyanka S. Numerade Educator ### Problem 76 Use differentiation to justify the formula$\int e^{x} d x=e^{x}+C$ Priyanka S. Numerade Educator ### Problem 77 Assuming that$x>0$, use differentiation to justify the formula$\int \frac{1}{x} d x=\ln |x|+C$ Priyanka S. Numerade Educator ### 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$s(x)=f(x)+g(x)$.] Priyanka S. Numerade Educator ### Problem 80 Show that the indefinite integral of the difference of two functions is the difference of the indefinite integrals. Priyanka S. 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?

Check back soon!

### 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.

Check back soon!

### 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.

### 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.

### 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.

### 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.

### 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? Priyanka S.

### 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? Priyanka S.

### 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.

### 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? Priyanka S.

### 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.
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. 