Chapter 7, Additional Topics in Integration Video Solutions, Applied Calculus for the MLSS A Brief Approach

Section 1

Integration by Parts

04:07

Problem 1

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

Naresh B.

Numerade Educator

01:30

Problem 2

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

Priyanka S.

Numerade Educator

04:25

Problem 3

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

Willis J.

Numerade Educator

02:47

Problem 4

Find indefinite integral.
$$\int 6 x e^{3 x} d x$$

Naresh B.

Numerade Educator

Problem 5

Find indefinite integral.
$$\int\left(e^{x}-x\right)^{2} d x$$

Check back soon!

07:51

Problem 6

Find indefinite integral.
$$\int\left(e^{-x}+x\right)^{2} d x$$

Willis J.

Numerade Educator

Problem 7

Find indefinite integral.
$$\int(x+1) e^{x} d x$$

Check back soon!

03:15

Problem 8

Find indefinite integral.
$$\int(x-3) e^{3 x} d x$$

Naresh B.

Numerade Educator

02:53

Problem 9

Find indefinite integral.
$$\int x(x+1)^{-3 / 2} d x$$

Naresh B.

Numerade Educator

03:12

Problem 10

Find indefinite integral.
$$\int x(x+4)^{-2} d x$$

Naresh B.

Numerade Educator

03:34

Problem 11

Find indefinite integral.
$$\int x \sqrt{x-5} d x$$

Naresh B.

Numerade Educator

04:14

Problem 12

Find indefinite integral.
$$\int \frac{3 x}{\sqrt{2 x+3}} d x$$

Naresh B.

Numerade Educator

03:50

Problem 13

Find indefinite integral.
$$\int x \ln 2 x d x$$

Naresh B.

Numerade Educator

03:39

Problem 14

Find indefinite integral.
$$\int x^{2} \ln 2 x d x$$

Naresh B.

Numerade Educator

03:31

Problem 15

Find indefinite integral.
$$\int x^{3} \ln x d x$$

Naresh B.

Numerade Educator

04:41

Problem 16

Find indefinite integral.
$$\int \sqrt{x} \ln x d x$$

Naresh B.

Numerade Educator

04:36

Problem 17

Find indefinite integral.
$$\int \sqrt{x} \ln \sqrt{x} d x$$

Naresh B.

Numerade Educator

Problem 18

Find indefinite integral.
$$\int \frac{\ln x}{\sqrt{x}} d x$$

Check back soon!

03:03

Problem 19

Find indefinite integral.
$$\int \frac{\ln x}{x^{2}} d x$$

Naresh B.

Numerade Educator

04:41

Problem 20

Find indefinite integral.
$$\int \frac{\ln x}{x^{3}} d x$$

Naresh B.

Numerade Educator

01:12

Problem 21

Find indefinite integral.
$$\int x e^{(x+1)^{2}} d x$$

Gregory H.

Numerade Educator

03:14

Problem 22

Find indefinite integral.
$$\int \ln \left(x e^{x^{2}}\right) d x$$

Willis J.

Numerade Educator

02:47

Problem 23

Find indefinite integral.
$\int \ln x d x$
Hint: Let $u=\ln x$ and $d v=d x$

Willis J.

Numerade Educator

03:24

Problem 24

Find indefinite integral.
$$\int \ln (x+1) d x$$

Naresh B.

Numerade Educator

02:40

Problem 25

Find indefinite integral.
$\int x^{2} e^{-x} d x$
Hint: Integrate by parts twice.

Steven C.

Numerade Educator

03:11

Problem 26

Find indefinite integral.
$\int e^{-\sqrt{x}} d x$
Hint: First, make the substitution $u=\sqrt{x}$; then integrate by parts.

Steven C.

Numerade Educator

02:52

Problem 27

Find indefinite integral.
$\int x(\ln x)^{2} d x$
Hint: Integrate by parts twice.

Steven C.

Numerade Educator

11:07

Problem 28

Find indefinite integral.
$-\int x \ln (x+1) d x$
Hint: First, make the substitution $u=x+1 ;$ then integrate by parts.

Willis J.

Numerade Educator

03:47

Problem 29

Evaluate each definite integral by using the method of integration by parts.
$$\int_{0}^{\ln 3} x e^{x} d x$$

Willis J.

Numerade Educator

03:06

Problem 30

Evaluate each definite integral by using the method of integration by parts.
$$\int_{0}^{2} x e^{-x} d x$$

Willis J.

Numerade Educator

04:31

Problem 31

Evaluate each definite integral by using the method of integration by parts.
$$\int_{1}^{4} 3 \ln x d x$$

Naresh B.

Numerade Educator

05:12

Problem 32

Evaluate each definite integral by using the method of integration by parts.
$$\int_{1}^{2} x \ln x d x$$

Naresh B.

Numerade Educator

05:03

Problem 33

Evaluate each definite integral by using the method of integration by parts.
$$\int_{0}^{2} x e^{2 x} d x$$

Naresh B.

Numerade Educator

07:40

Problem 34

Evaluate each definite integral by using the method of integration by parts.
$$\int_{0}^{1} x^{2} e^{-x} d x$$

Naresh B.

Numerade Educator

06:53

Problem 35

Find the function $f$ given that the slope of the tangent line to the graph of $f$ at any point $(x, f(x))$ is $x e^{-2 x}$ and that the graph passes through the point (0,3).

Naresh B.

Numerade Educator

07:16

Problem 36

Find the function $f$ given that the slope of the tangent line to the graph of $f$ at any point $f$ at any point $(x, f(x))$ is $x \sqrt{x+1}$ and that the graph passes through the point (3,6).

Naresh B.

Numerade Educator

05:32

Problem 37

Find the area of the region under the graph of $f(x)=\ln x$ from $x=1$ to $x=5$.

Naresh B.

Numerade Educator

05:17

Problem 38

Find the area of the region under the graph of $f(x)=x e^{-x}$ from $x=0$ to $x=3$.

Naresh B.

Numerade Educator

09:28

Problem 39

Velocity of a Dragster The velocity of a dragster $t$ sec after leaving the starting line is
$$100 t e^{-0.2 t}$$
ft/sec. What is the distance covered by the dragster in the first 10 sec of its run?

Naresh B.

Numerade Educator

Production of Steam Coal In keeping with the projected increase in worldwide demand for steam coal, the boilerfiring fuel used for generating electricity, the management of Consolidated Mining has decided to step up its mining operations. Plans call for increasing the yearly production of steam coal over its current level by
$$2 t e^{-0.05 t}$$
million metric tons/year for the next 20 years. The current yearly production is 20 million metric tons. Find a function that describes Consolidated's total production of steam coal at the end of $t$ years. How much coal will Consolidated have produced over the next 20 years if this plan is carried out?

Check back soon!

08:01

Problem 41

Compact Disc Sales Sales of the latest recording by Brittania, a British rock group, are currently $2 t e^{-0.1 t}$ units/week (each unit representing $10,000 \mathrm{CDs}$ ), where $t$ denotes the number of weeks since the recording's release. Find an expression that gives the total number of CDs sold as a function of $t.$

Naresh B.

Numerade Educator

09:54

Problem 42

Concentration of a Drug in the Bloodstream The concentration (in milligrams per milliliter) of a certain drug in a patient's bloodstream $t$ hr after it has been administered is given by $C(t)=3 t e^{-t / 3} \mathrm{mg} / \mathrm{mL} .$ Find the average concentration of the drug in the patient's bloodstream over the first 12 hr after administration.

Naresh B.

Numerade Educator

Problem 43

Alcohol-Related Traffic Accidents As a result of increasingly stiff laws aimed at reducing the number of alcoholrelated traffic accidents in a certain state, preliminary data indicate that the number of such accidents has been changing at the rate of
$$R(t)=-10-t e^{0.1 t}$$
accidents/month $t$ months after the laws took effect. There were 982 alcohol-related accidents for the year before the enactment of the laws. Determine how many alcoholrelated accidents were expected during the first year the laws were in effect.

Check back soon!

10:50

Problem 44

Average Price of a Commodity The price of a certain commodity in dollars per unit at time $t$ (measured in weeks) is given by
$$p=8+4 e^{-2 t}+t e^{-2 t}$$
What is the average price of the commodity over the 4 -week period from $t=0$ to $t=4 ?$

Naresh B.

Numerade Educator

Problem 45

Growth of HMOs The membership of the Cambridge Community Health Plan (a health maintenance organization) is projected to grow at the rate of
$$9 \sqrt{t+1} \ln \sqrt{t+1}$$
thousand people/year, $t$ years from now. If the HMO's current membership is 50,000 , what will be the membership 5 years from now?

Check back soon!

11:48

Problem 46

Distance Covered by a Car The velocity of the Zephyr electric sedan traveling along a straight road is
$$v(t)=8 t \ln (t+1) \quad(0 \leq t \leq 6)$$
ft/sec $t$ sec after starting from rest. What is the total distance covered by the car at the end of 6 sec?

Willis J.

Numerade Educator

08:30

Problem 47

Credit Card Losses in the United Kingdom The mail and the Internet are major routes for fraud against merchants who sell and ship products. These merchants must often accept credit cards that are not physically present (called CNPs, "credit cards not present"). The amount of CNP fraud in the United Kingdom from 2008 through 2012 was
$$f(t)=328.9-92.07 \ln (t+1) \quad(0 \leq t \leq 4)$$
million British pounds (GBP) in year $t,$ where $t=0$ corresponds to $2008 .$ What was the average amount of CNP fraud losses in the United Kingdom over the period from 2008 through 2012 ?

Willis J.

Numerade Educator

Problem 48

U.S. Smartphone Users Smartphone users, as a percentage of the population, are projected to be
$$f(t)=18.952+14.088 \ln (t+1) \quad(0 \leq t \leq 5)$$
percent in year $t,$ where $t$ is measured in years, with $t=0$ corresponding to $2010 .$ What is the projected average percentage of the population that will be smartphone users over the period from 2010 through 2015?

Check back soon!

Problem 49

Future Value of an Income Stream The chef-owner of the Vegan Restaurant expects that in the first 5 years of its operation, revenue will be generated at the rate of $R(t)=100+20 t$ thousand dollars/year, where $t$ is the number of years after the grand opening of the restaurant. If the prevailing rate of interest for the period is $5 \% /$ year compounded continuously, what will be the value of the restaurant's revenue stream at the end of 5 years?

Check back soon!

Problem 50

Future Value of an Investment Laura, the sole proprietor of Laura's Boutique, plans to invest $R(t)=10,000 e^{-0.02 t}$ dollars/year into her savings account for 5 years, starting now. If the bank pays interest at the rate of $3 \% /$ year compounded continuously, how much will Laura have in her account at the end of the 5-year period?

Check back soon!

Problem 51

Rate of Return on an Investment Suppose an investment is expected to generate income at the rate of
$$P(t)=30,000+800 t$$
dollars/year for the next 5 years. Find the present value of the income from this investment if the prevailing interest rate is $5 \% /$ year compounded continuously.
Hint: Use Formula ( 19 ). Section $6.7 .$

Check back soon!

Problem 52

Present Value of a Franchise Tracy purchased a 15 -year franchise for a computer outlet store that is expected to generate income at the rate of
$$P(t)=50,000+3000 t$$
dollars/year. If the prevailing interest rate is $4 \% /$ year compounded continuously, find the present value of the income from franchise.

Check back soon!

Problem 53

Consumers' Surplus for Inflatable Canoes Refer to Section $6.7 .$ The demand function for Sea Hawk inflatable canoes is given by
$$p=300-2(x+1) \ln (x+1) \quad(x \geq 0)$$
where $p$ is the unit price in dollars and $x$ (in units of a hundred) is the quantity demanded. Estimate the consumers' surplus if the quantity demanded is set at 2000 units.

Check back soon!

Problem 54

Producers' Surplus for Laptop Locks Refer to Section 6.7. The supply function for a certain brand of laptop lock is given by
$$p=x \ln \left(0.1 x^{2}+1\right)+5 \quad(0 \leq x \leq 10)$$
where $p$ is the unit price in dollars and $x$ is the corresponding quantity that the manufacturers will make available in the market, measured in units of a thousand. Find the producers' surplus if the quantity demanded is 7000.

Check back soon!

Problem 55

Gini Index Refer to Section $6.7 .$ The income distribution of a country is described by the Lorenz function
$$f(x)=x e^{2(x-1)} \qquad(0 \leq x \leq 1)$$
Find the Gini Index for the income distribution of this country.

Check back soon!

Problem 56

Diffusion A cylindrical membrane with inner radius $r_{1}$ cm and outer radius $r_{2}$ cm containing a chemical solution is introduced into a salt bath with constant concentration $c_{2}$ moles/L (see the accompanying figure).
If the concentration of the chemical inside the membrane is kept constant at a different concentration of $c_{1}$ moles/L, then the concentration of the chemical across the membrane will be given by
$$c(r)=\left(\frac{c_{1}-c_{2}}{\ln r_{1}-\ln r_{2}}\right)\left(\ln r-\ln r_{2}\right)+c_{2} \quad\left(r_{1}< r< r_{2}\right)$$
moles/L. Find the average concentration of the chemical across the membrane from $r=r_{1}$ to $r=r_{2}.$

Check back soon!

Problem 57

A Mixture Problem Two tanks are connected in tandem as shown in the following figure. Each tank contains 60 gal of water. Starting at time $t=0,$ brine containing 3 ib/gal of salt flows into tank 1 at the rate of 2 gal/min. The mixture then enters and leaves tank 2 at the same rate. The mixtures in both tanks are stirred uniformly. It can be shown that the amount of salt in tank 2 after $t$ min is given by
$$A(t)=180\left(1-e^{-t / 30}\right)-6 t e^{-t / 30}$$
where $A(t)$ is measured in pounds.
a. What is the initial amount of salt in tank $2 ?$
b. What is the amount of salt in tank 2 after $3 \mathrm{hr}(180 \mathrm{min}) ?$
c. What is the average amount of salt in tank 2 over the first $3 \mathrm{hr}$ ?

Check back soon!

05:27

Problem 58

Suppose $f^{\prime \prime}$ is continuous on [1,3] and $f(1)=2, f(3)=-1, f^{\prime}(1)=2,$ and $f^{\prime}(3)=-1 .$ Evaluate $\int_{1}^{3} x f^{\prime \prime}(x) d x.$

Naresh B.

Numerade Educator

01:25

Problem 59

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false.
$$\int u d v+\int v d u=u v$$

Naresh B.

Numerade Educator

02:48

Problem 60

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false.
$$\int e^{x} g^{\prime}(x) d x=e^{x} g(x)-\int e^{x} g(x) d x$$

Willis J.

Numerade Educator

02:15

Problem 61

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false.
$$\int u v d w=u v w-\int u w d v-\int v w d u$$

Naresh B.

Numerade Educator

01:53

Problem 62

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false.
$$\int_{a}^{b} u d v=u v-\int_{a}^{b} v d u$$

Naresh B.

Numerade Educator