Calculus an Applied Approach

Ron Larson, David C, Falvo

Chapter 6

Techniques of Integration - all with Video Answers

Educators

Section 1

Integration by Parts and Present Value

Problem 1

identify $u$ and $d v$ for finding the integral using integration by parts. (Do not evaluate the integral.)
$$
\int x e^{3 x} d x
$$

Vikash Ranjan

Vikash Ranjan

Numerade Educator

Problem 2

identify $u$ and $d v$ for finding the integral using integration by parts. (Do not evaluate the integral.)
$$
\int x^{2} e^{3 x} d x
$$

Vikash Ranjan

Numerade Educator

Problem 3

identify $u$ and $d v$ for finding the integral using integration by parts. (Do not evaluate the integral.)
$$
\int x \ln 2 x d x
$$

Vikash Ranjan

Numerade Educator

Problem 4

identify $u$ and $d v$ for finding the integral using integration by parts. (Do not evaluate the integral.)
$$
\int \ln 4 x d x
$$

Vikash Ranjan

Numerade Educator

Problem 5

use integration by parts to find the indefinite integral.
$$
\int x e^{3 x} d x
$$

Vikash Ranjan

Numerade Educator

Problem 6

use integration by parts to find the indefinite integral.
$$
\int x e^{-x} d x
$$

Vikash Ranjan

Numerade Educator

Problem 7

use integration by parts to find the indefinite integral.
$$
\int x^{2} e^{-x} d x
$$

Vikash Ranjan

Numerade Educator

Problem 8

use integration by parts to find the indefinite integral.
$$
\int x^{2} e^{2 x} d x
$$

Vikash Ranjan

Numerade Educator

Problem 9

use integration by parts to find the indefinite integral.
$$
\int \ln 2 x d x
$$

Vikash Ranjan

Numerade Educator

Problem 10

use integration by parts to find the indefinite integral.
$$
\int \ln x^{2} d x
$$

Vikash Ranjan

Numerade Educator

Problem 11

find the indefinite integral. (Hint: Integration by parts is not required for all the integrals.)
$$
\int e^{4 x} d x
$$

Vikash Ranjan

Numerade Educator

Problem 12

find the indefinite integral. (Hint: Integration by parts is not required for all the integrals.)
$$
\int e^{-2 x} d x
$$

Vikash Ranjan

Numerade Educator

Problem 13

find the indefinite integral. (Hint: Integration by parts is not required for all the integrals.)
$$
\int x e^{4 x} d x
$$

Vikash Ranjan

Numerade Educator

Problem 14

find the indefinite integral. (Hint: Integration by parts is not required for all the integrals.)
$$
\int x e^{-2 x} d x
$$

Vikash Ranjan

Numerade Educator

Problem 15

find the indefinite integral. (Hint: Integration by parts is not required for all the integrals.)
$$
\int x e^{x^{2}} d x
$$

Vikash Ranjan

Numerade Educator

Problem 16

find the indefinite integral. (Hint: Integration by parts is not required for all the integrals.)
$$
\int x^{2} e^{x^{3}} d x
$$

Vikash Ranjan

Numerade Educator

Problem 17

find the indefinite integral. (Hint: Integration by parts is not required for all the integrals.)
$$
\int \frac{x}{e^{x}} d x
$$

Vikash Ranjan

Numerade Educator

Problem 18

find the indefinite integral. (Hint: Integration by parts is not required for all the integrals.)
$$
\int \frac{2 x}{e^{x}} d x
$$

Vikash Ranjan

Numerade Educator

Problem 19

find the indefinite integral. (Hint: Integration by parts is not required for all the integrals.)
$$
\int 2 x^{2} e^{x} d x
$$

Vikash Ranjan

Numerade Educator

Problem 20

find the indefinite integral. (Hint: Integration by parts is not required for all the integrals.)
$$
\int \frac{1}{2} x^{3} e^{x} d x
$$

Vikash Ranjan

Numerade Educator

Problem 21

find the indefinite integral. (Hint: Integration by parts is not required for all the integrals.)
$$
\int t \ln (t+1) d t
$$

Vikash Ranjan

Numerade Educator

Problem 22

find the indefinite integral. (Hint: Integration by parts is not required for all the integrals.)
$$
\int x^{3} \ln x d x
$$

Vikash Ranjan

Numerade Educator

Problem 23

find the indefinite integral. (Hint: Integration by parts is not required for all the integrals.)
$$
\int(x-1) e^{x} d x
$$

Vikash Ranjan

Numerade Educator

Problem 24

find the indefinite integral. (Hint: Integration by parts is not required for all the integrals.)
$$
\int x^{4} \ln x d x
$$

Vikash Ranjan

Numerade Educator

Problem 25

find the indefinite integral. (Hint: Integration by parts is not required for all the integrals.)
$$
\int \frac{e^{1 / t}}{t^{2}} d t
$$

Vikash Ranjan

Numerade Educator

Problem 26

find the indefinite integral. (Hint: Integration by parts is not required for all the integrals.)
$$
\int \frac{1}{x(\ln x)^{3}} d x
$$

Vikash Ranjan

Numerade Educator

Problem 27

find the indefinite integral. (Hint: Integration by parts is not required for all the integrals.)
$$
\int x(\ln x)^{2} d x
$$

Vikash Ranjan

Numerade Educator

Problem 28

find the indefinite integral. (Hint: Integration by parts is not required for all the integrals.)
$$
\int \ln 3 x d x
$$

Vikash Ranjan

Numerade Educator

Problem 29

find the indefinite integral. (Hint: Integration by parts is not required for all the integrals.)
$$
\int \frac{(\ln x)^{2}}{x} d x
$$

Vikash Ranjan

Numerade Educator

Problem 30

find the indefinite integral. (Hint: Integration by parts is not required for all the integrals.)
$$
\int \frac{1}{x \ln x} d x
$$

Vikash Ranjan

Numerade Educator

Problem 31

find the indefinite integral. (Hint: Integration by parts is not required for all the integrals.)
$$
\int \frac{\ln x}{x^{2}} d x
$$

Vikash Ranjan

Numerade Educator

Problem 32

find the indefinite integral. (Hint: Integration by parts is not required for all the integrals.)
$$
\int \frac{\ln 2 x}{x^{2}} d x
$$

Vikash Ranjan

Numerade Educator

Problem 33

find the indefinite integral. (Hint: Integration by parts is not required for all the integrals.)
$$
\int x \sqrt{x-1} d x
$$

Vikash Ranjan

Numerade Educator

Problem 34

find the indefinite integral. (Hint: Integration by parts is not required for all the integrals.)
$$
\int \frac{x}{\sqrt{x-1}} d x
$$

Vikash Ranjan

Numerade Educator

Problem 35

find the indefinite integral. (Hint: Integration by parts is not required for all the integrals.)
$$
\int x(x+1)^{2} d x
$$

Vikash Ranjan

Numerade Educator

Problem 36

find the indefinite integral. (Hint: Integration by parts is not required for all the integrals.)
$$
\int \frac{x}{\sqrt{2+3 x}} d x
$$

Vikash Ranjan

Numerade Educator

Problem 37

find the indefinite integral. (Hint: Integration by parts is not required for all the integrals.)
$$
\int \frac{x e^{2 x}}{(2 x+1)^{2}} d x
$$

Vikash Ranjan

Numerade Educator

Problem 38

find the indefinite integral. (Hint: Integration by parts is not required for all the integrals.)
$$
\int \frac{x^{3} e^{x^{2}}}{\left(x^{2}+1\right)^{2}} d x
$$

Vikash Ranjan

Numerade Educator

Problem 39

evaluate the definite integral.
$$
\int_{1}^{2} x^{2} e^{x} d x
$$

Vikash Ranjan

Numerade Educator

Problem 40

evaluate the definite integral.
$$
\int_{0}^{2} \frac{x^{2}}{e^{x}} d x
$$

Vikash Ranjan

Numerade Educator

Problem 41

evaluate the definite integral.
$$
\int_{0}^{4} \frac{x}{e^{x / 2}} d x
$$

Vikash Ranjan

Numerade Educator

Problem 42

evaluate the definite integral.
$$
\int_{1}^{2} x^{2} \ln x d x
$$

Vikash Ranjan

Numerade Educator

Problem 43

evaluate the definite integral.
$$
\int_{1}^{e} x^{5} \ln x d x
$$

Vikash Ranjan

Numerade Educator

Problem 44

evaluate the definite integral.
$$
\int_{1}^{e} 2 x \ln x d x
$$

Vikash Ranjan

Numerade Educator

Problem 45

evaluate the definite integral.
$$
\int_{-1}^{0} \ln (x+2) d x
$$

Vikash Ranjan

Numerade Educator

Problem 46

evaluate the definite integral.
$$
\int_{0}^{1} \ln (1+2 x) d x
$$

Vikash Ranjan

Numerade Educator

Problem 47

find the area of the region bounded by the graphs of the equations. Then use a graphing utility to graph the region and verify your answer.
$$
y=x^{3} e^{x}, y=0, x=0, x=2
$$

Vikash Ranjan

Numerade Educator

Problem 48

find the area of the region bounded by the graphs of the equations. Then use a graphing utility to graph the region and verify your answer.
$$
y=\left(x^{2}-1\right) e^{x}, y=0, x=-1, x=1
$$

Vikash Ranjan

Numerade Educator

Problem 49

find the area of the region bounded by the graphs of the equations. Then use a graphing utility to graph the region and verify your answer.
$$
y=x^{2} \ln x, y=0, x=1, x=e
$$

Vikash Ranjan

Numerade Educator

Problem 50

find the area of the region bounded by the graphs of the equations. Then use a graphing utility to graph the region and verify your answer.
$$
y=\frac{\ln x}{x^{2}}, y=0, x=1, x=e
$$

Vikash Ranjan

Numerade Educator

Problem 51

use integration by parts to verify the formula.
$$
\begin{array}{l}{\int x^{n} \ln x d x=\frac{x^{n+1}}{(n+1)^{2}}[-1+(n+1) \ln x]+C} \\ {n \neq-1}\end{array}
$$

Vikash Ranjan

Numerade Educator

Problem 52

use integration by parts to verify the formula.
$$
\int x^{n} e^{a x} d x=\frac{x^{n} e^{a x}}{a}-\frac{n}{a} \int x^{n-1} e^{a x} d x, \quad n>0
$$

Vikash Ranjan

Numerade Educator

Problem 53

use the results of Exercises 51 and 52 to find the indefinite integral.
$$
\int x^{2} e^{5 x} d x
$$

Vikash Ranjan

Numerade Educator

Problem 54

use the results of Exercises 51 and 52 to find the indefinite integral.
$$
\int x e^{-3 x} d x
$$

Vikash Ranjan

Numerade Educator

Problem 55

use the results of Exercises 51 and 52 to find the indefinite integral.
$$
\int x^{-2} \ln x d x
$$

Vikash Ranjan

Numerade Educator

Problem 56

use the results of Exercises 51 and 52 to find the indefinite integral.
$$
\int x^{1 / 2} \ln x d x
$$

Vikash Ranjan

Numerade Educator

Problem 57

find the area of the region bounded by the graphs of the given equations.
$$
\begin{array}{l}{y=x e^{-x}} \\ {y=0, x=4}\end{array}
$$

Vikash Ranjan

Numerade Educator

Problem 58

find the area of the region bounded by the graphs of the given equations.
$$
\begin{array}{l}{y=\frac{1}{9} x e^{-x / 3}} \\ {y=0, x=0, x=3}\end{array}
$$

Vikash Ranjan

Numerade Educator

Problem 59

find the area of the region bounded by the graphs of the given equations.
$$
\begin{array}{l}{y=x \ln x} \\ {y=0, x=e}\end{array}
$$

Vikash Ranjan

Numerade Educator

Problem 60

find the area of the region bounded by the graphs of the given equations.
$$
\begin{array}{l}{y=x^{-3} \ln x} \\ {y=0, x=e}\end{array}
$$

Vikash Ranjan

Numerade Educator

Problem 61

use a symbolic integration utility to evaluate the integral.
$$
\int_{0}^{2} t^{3} e^{-4 t} d t
$$

Vikash Ranjan

Numerade Educator

Problem 62

use a symbolic integration utility to evaluate the integral.
$$
\int_{1}^{4} \ln x\left(x^{2}+4\right) d x
$$

Vikash Ranjan

Numerade Educator

Problem 63

use a symbolic integration utility to evaluate the integral.
$$
\int_{0}^{5} x^{4}\left(25-x^{2}\right)^{3 / 2} d x
$$

Vikash Ranjan

Numerade Educator

Problem 64

use a symbolic integration utility to evaluate the integral.
$$
\int_{1}^{e} x^{9} \ln x d x
$$

Vikash Ranjan

Numerade Educator

Problem 65

Demand A manufacturing company forecasts that the demand $x$ (in units per year) for its product over the next 10 years can be modeled by $x=500\left(20+t e^{-0.1 t}\right)$ for
$0 \leq t \leq 10,$ where $t$ is the time in years.
(a) Use a graphing utility to decide whether the company is forecasting an increase or a decrease in demand over the decade.
(b) According to the model, what is the total demand over the next 10 years?
(c) Find the average annual demand during the 10 -year period.

Joanie Morris

Numerade Educator

Problem 66

Capital Campaign The board of trustees of a college is planning a five-year capital gifts campaign to raise money for the college. The goal is to have an annual gift income $I$ that is modeled by $I=2000\left(375+68 t e^{-0.2 t}\right)$ for $0 \leq t \leq 5,$ where $t$ is the time in years.
(a) Use a graphing utility to decide whether the board of trustees expects the gift income to increase or decrease over the five-year period.
(b) Find the expected total gift income over the five-year period.
(c) Determine the average annual gift income over the five-year period. Compare the result with the income given when $t=3 .$

Julian Wong

Numerade Educator

Problem 67

Memory Model A model for the ability $M$ of a child to memorize, measured on a scale from 0 to 10 , is $M=1+1.6 t \ln t, \quad 0<t \leq 4$ where $t$ is the child's age in years. Find the average value of this model between
(a) the child's first and second birthdays.
(b) the child's third and fourth birthdays.

Adrian Co

Numerade Educator

Problem 68

Revenue A company sells a seasonal product. The revenue $R$ (in dollars per year) generated by sales of the product can be modeled by $R=410.5 t^{2} e^{-t / 30}+25,000, \quad 0 \leq t \leq 365$ where $t$ is the time in days.
(a) Find the average daily receipts during the first quarter, which is given by $0 \leq t \leq 90$.
(b) Find the average daily receipts during the fourth quarter, which is given by $274 \leq t \leq 365$.
(c) Find the total daily receipts during the year.

James Kiss

Numerade Educator

Problem 69

Present Value In Exercises $69-74,$ find the present value of the income $c$ (measured in dollars) over $t_{1}$ years at the given annual inflation rate $r .$
$$
c=5000, r=4 \%, t_{1}=4 \text { years }
$$

Adrian Co

Numerade Educator

Problem 70

Present Value In Exercises $69-74,$ find the present value of the income $c$ (measured in dollars) over $t_{1}$ years at the given annual inflation rate $r .$
$$
c=450, r=4 \%, t_{1}=10 \text { years }
$$

Adrian Co

Numerade Educator

Problem 71

Present Value In Exercises $69-74,$ find the present value of the income $c$ (measured in dollars) over $t_{1}$ years at the given annual inflation rate $r .$
$$
c=100,000+4000 t, r=5 \%, t_{1}=10 \text { years }
$$

Adrian Co

Numerade Educator

Problem 72

Present Value In Exercises $69-74,$ find the present value of the income $c$ (measured in dollars) over $t_{1}$ years at the given annual inflation rate $r .$
$$
c=30,000+500 t, r=7 \%, t_{1}=6 \text { years }
$$

Adrian Co

Numerade Educator

Problem 73

Present Value In Exercises $69-74,$ find the present value of the income $c$ (measured in dollars) over $t_{1}$ years at the given annual inflation rate $r .$
$$
c=1000+50 e^{t / 2}, r=6 \%, t_{1}=4 \text { years }
$$

Adrian Co

Numerade Educator

Problem 74

Present Value In Exercises $69-74,$ find the present value of the income $c$ (measured in dollars) over $t_{1}$ years at the given annual inflation rate $r .$
$$
c=5000+25 t e^{t / 10}, r=6 \%, t_{1}=10 \text { years }
$$

Adrian Co

Numerade Educator

Problem 75

Present Value A company expects its income $c$ during the next 4 years to be modeled by $c=150,000+75,000 t$
(a) Find the actual income for the business over the 4 years.
(b) Assuming an annual inflation rate of $4 \%,$ what is the present value of this income?

Adrian Co

Numerade Educator

Problem 76

Present Value A professional athlete signs a three-year contract in which the earnings can be modeled by $c=300,000+125,000 t$
(a) Find the actual value of the athlete's contract.
(b) Assuming an annual inflation rate of $3 \%,$ what is the present value of the contract?

Adrian Co

Numerade Educator

Problem 77

Future Value In Exercises 77 and $78,$ find the future value of the income (in dollars) given by $f(t)$ over $t_{1}$ years at the annual interest rate of $r$. If the function $f_{1}$ represents a continuous investment over a period of $t_{1}$ years at an annual interest rate of $r$ (compounded continuously), then the future value of the investment is given by Future value $=e^{r t_{1}} \int_{0}^{t_{1}} f(t) e^{-r t} d t$
$$
f(t)=3000, r=8 \%, t_{1}=10 \text { years }
$$

Vikash Ranjan

Numerade Educator

Problem 78

Future Value In Exercises 77 and $78,$ find the future value of the income (in dollars) given by $f(t)$ over $t_{1}$ years at the annual interest rate of $r$. If the function $f_{1}$ represents a continuous investment over a period of $t_{1}$ years at an annual interest rate of $r$ (compounded continuously), then the future value of the investment is given by Future value $=e^{r t_{1}} \int_{0}^{t_{1}} f(t) e^{-r t} d t$
$$
f(t)=3000 e^{0.05 t}, r=10 \%, t_{1}=5 \text { years }
$$

Vikash Ranjan

Numerade Educator

Problem 79

Use the equation from Exercises 77 and 78 to calculate the following.
(a) The future value of $\$ 1200$ saved each year for 10 years earning $7 \%$ interest.
(b) A person who wishes to invest $\$ 1200$ each year finds one investment choice that is expected to pay $9 \%$ interest per year and another, riskier choice that may pay $10 \%$ interest per year. What is the difference in return (future value) if the investment is made for
15 years?

Vikash Ranjan

Numerade Educator

Problem 80

In 2006 the total cost of attending Pennsylvania State University for 1 year was estimated to be $\$ 20,924$. Assume your grandparents had continuously invested in a college fund according to the model $f(t)=400 t$ for 18 years, at an annual interest rate of $10 \% .$ Will the fund
have grown enough to allow you to cover 4 years. of expenses at Pennsylvania State University?

Amit Srivastava

Amit Srivastava

Numerade Educator

Problem 81

Use a program similar to the Midpoint Rule program on page 404 with $n=10$ to approximate
$$\int_{1}^{4} \frac{4}{\sqrt{x}+\sqrt[3]{x}} d x$$

Joseph David

Numerade Educator

Problem 82

Use a program similar to the Midpoint Rule program on page 404 with $n=12$ to approximate the area of the region bounded by the graphs of $y=\frac{10}{\sqrt{x} e^{x}}, y=0, x=1,$ and $x=4$

Mohamed Raafat Mohamed

Mohamed Raafat Mohamed

Numerade Educator