Calculus of a Single Variable

Ron Larson, Bruce Edwards

Chapter 8

Integration Techniques, L'Hopital's Rule, and Improper Integrals - all with Video Answers

Educators

Section 1

Basic Integration Rules

Problem 1

Choosing an Antiderivative In Exercises $1-4,$ select the correct antiderivative.
$\frac{d y}{d x}=\frac{x}{\sqrt{x^{2}+1}}$
$$
\begin{array}{ll}{\text { (a) } 2 \sqrt{x^{2}+1}+C} & {\text { (b) } \sqrt{x^{2}+1}+C} \\ {\text { (c) } \frac{1}{2} \sqrt{x^{2}+1}+C} & {\text { (d) } \ln \left(x^{2}+1\right)+C}\end{array}
$$

Christy Galilei

Christy Galilei

Numerade Educator

Problem 2

Choosing an Antiderivative In Exercises $1-4,$ select the correct antiderivative.
$\frac{d y}{d x}=\frac{x}{x^{2}+1}$
$$
\begin{array}{ll}{\text { (a) } \ln \sqrt{x^{2}+1}+c} & {\text { (b) } \frac{2 x}{\left(x^{2}+1\right)^{2}}+c} \\ {\text { (c) arctan } x+c} & {\text { (d) } \ln \left(x^{2}+1\right)+c}\end{array}
$$

Christy Galilei

Christy Galilei

Numerade Educator

Problem 3

Choosing an Antiderivative In Exercises $1-4,$ select the correct antiderivative.
$\frac{d y}{d x}=\frac{1}{x^{2}+1}$
$$
\begin{array}{ll}{\text { (a) } \ln \sqrt{x^{2}+1}+C} & {\text { (b) } \frac{2 x}{\left(x^{2}+1\right)^{2}}+C} \\ {\text { (c) } \arctan x+c} & {\text { (d) } \ln \left(x^{2}+1\right)+c}\end{array}
$$

Ashley Boni

Numerade Educator

Problem 4

Choosing an Antiderivative In Exercises $1-4,$ select the correct antiderivative.
$\frac{d y}{d x}=x \cos \left(x^{2}+1\right)$
$$
\begin{array}{ll}{\text { (a) } 2 x \sin \left(x^{2}+1\right)+C} & {\text { (b) }-\frac{1}{2} \sin \left(x^{2}+1\right)+C} \\ {\text { (c) } \frac{1}{2} \sin \left(x^{2}+1\right)+C} & {\text { (d) }-2 x \sin \left(x^{2}+1\right)+C}\end{array}
$$

Christy Galilei

Christy Galilei

Numerade Educator

Problem 5

Choosing a Formula In Exercises $5-14$ , select the basic integration formula you can use to find the integral, and identify $u$ and $a$ when appropriate.
$$
\int(5 x-3)^{4} d x
$$

Harriet O'Brien

Harriet O'Brien

Numerade Educator

Problem 6

Choosing a Formula In Exercises $5-14$ , select the basic integration formula you can use to find the integral, and identify $u$ and $a$ when appropriate.
$$
\int \frac{2 t+1}{t^{2}+t-4} d t
$$

Ashley Boni

Numerade Educator

Problem 7

Choosing a Formula In Exercises $5-14$ , select the basic integration formula you can use to find the integral, and identify $u$ and $a$ when appropriate.
$$
\int \frac{1}{\sqrt{x}(1-2 \sqrt{x})} d x
$$

Harriet O'Brien

Harriet O'Brien

Numerade Educator

Problem 8

Choosing a Formula In Exercises $5-14$ , select the basic integration formula you can use to find the integral, and identify $u$ and $a$ when appropriate.
$$
\int \frac{2}{(2 t-1)^{2}+4} d t
$$

Ashley Boni

Numerade Educator

Problem 9

Choosing a Formula In Exercises $5-14$ , select the basic integration formula you can use to find the integral, and identify $u$ and $a$ when appropriate.
$$
\int \frac{3}{\sqrt{1-t^{2}}} d t
$$

Pravakar Paul

Numerade Educator

Problem 10

Choosing a Formula In Exercises $5-14$ , select the basic integration formula you can use to find the integral, and identify $u$ and $a$ when appropriate.
$$
\int \frac{-2 x}{\sqrt{x^{2}-4}} d x
$$

Ashley Boni

Numerade Educator

Problem 11

Choosing a Formula In Exercises $5-14$ , select the basic integration formula you can use to find the integral, and identify $u$ and $a$ when appropriate.
$$
\int t \sin t^{2} d t
$$

Pravakar Paul

Numerade Educator

Problem 12

Choosing a Formula In Exercises $5-14$ , select the basic integration formula you can use to find the integral, and identify $u$ and $a$ when appropriate.
$$
\int \sec 5 x \tan 5 x d x
$$

Ashley Boni

Numerade Educator

Problem 13

Choosing a Formula In Exercises $5-14$ , select the basic integration formula you can use to find the integral, and identify $u$ and $a$ when appropriate.
$$
\int(\cos x) e^{\sin x} d x
$$

Pravakar Paul

Numerade Educator

Problem 14

Choosing a Formula In Exercises $5-14$ , select the basic integration formula you can use to find the integral, and identify $u$ and $a$ when appropriate.
$$
\int \frac{1}{x \sqrt{x^{2}-4}} d x
$$

Ashley Boni

Numerade Educator

Problem 15

Finding an Indefinite Integral In Exercises $15-46$ , find the indefinite integral.
$$
\int 14(x-5)^{6} d x
$$

Charles Carter

Numerade Educator

Problem 16

Finding an Indefinite Integral In Exercises $15-46$ , find the indefinite integral.
$$
\int \frac{5}{(t+6)^{3}} d t
$$

Ashley Boni

Numerade Educator

Problem 17

Finding an Indefinite Integral In Exercises $15-46$ , find the indefinite integral.
$$
\int \frac{7}{(z-10)^{7}} d z
$$

Kay Macdonald

Numerade Educator

Problem 18

Finding an Indefinite Integral In Exercises $15-46$ , find the indefinite integral.
$$
\int t^{3} \sqrt{t^{4}+1} d t
$$

Ashley Boni

Numerade Educator

Problem 19

Finding an Indefinite Integral In Exercises $15-46$ , find the indefinite integral.
$$
\int\left[v+\frac{1}{(3 v-1)^{3}}\right] d v
$$

Christy Galilei

Christy Galilei

Numerade Educator

Problem 20

Finding an Indefinite Integral In Exercises $15-46$ , find the indefinite integral.
$$
\int\left[4 x-\frac{2}{(2 x+3)^{2}}\right] d x
$$

Ashley Boni

Numerade Educator

Problem 21

Finding an Indefinite Integral In Exercises $15-46$ , find the indefinite integral.
$$
\int \frac{t^{2}-3}{-t^{3}+9 t+1} d t
$$

Pravakar Paul

Numerade Educator

Problem 22

Finding an Indefinite Integral In Exercises $15-46$ , find the indefinite integral.
$$
\int \frac{x+1}{\sqrt{3 x^{2}+6 x}} d x
$$

Ashley Boni

Numerade Educator

Problem 23

Finding an Indefinite Integral In Exercises $15-46$ , find the indefinite integral.
$$
\int \frac{x^{2}}{x-1} d x
$$

Pravakar Paul

Numerade Educator

Problem 24

Finding an Indefinite Integral In Exercises $15-46$ , find the indefinite integral.
$$
\int \frac{3 x}{x+4} d x
$$

Ashley Boni

Numerade Educator

Problem 25

Finding an Indefinite Integral In Exercises $15-46$ , find the indefinite integral.
$$
\int \frac{e^{x}}{1+e^{x}} d x
$$

Christy Galilei

Christy Galilei

Numerade Educator

Problem 26

Finding an Indefinite Integral In Exercises $15-46$ , find the indefinite integral.
$$
\int\left(\frac{1}{2 x+5}-\frac{1}{2 x-5}\right) d x
$$

Christy Galilei

Christy Galilei

Numerade Educator

Problem 27

Finding an Indefinite Integral In Exercises $15-46$ , find the indefinite integral.
$$
\int\left(5+4 x^{2}\right)^{2} d x
$$

DW

Dimple Waghela

Numerade Educator

Problem 28

Finding an Indefinite Integral In Exercises $15-46$ , find the indefinite integral.
$$
\int x\left(3+\frac{2}{x}\right)^{2} d x
$$

Ashley Boni

Numerade Educator

Problem 29

Finding an Indefinite Integral In Exercises $15-46$ , find the indefinite integral.
$$
\int x \cos 2 \pi x^{2} d x
$$

Pravakar Paul

Numerade Educator

Problem 30

Finding an Indefinite Integral In Exercises $15-46$ , find the indefinite integral.
$$
\int \csc \pi x \cot \pi x d x
$$

Ashley Boni

Numerade Educator

Problem 31

Finding an Indefinite Integral In Exercises $15-46$ , find the indefinite integral.
$$
\int \frac{\sin x}{\sqrt{\cos x}} d x
$$

Pravakar Paul

Numerade Educator

Problem 32

Finding an Indefinite Integral In Exercises $15-46$ , find the indefinite integral.
$$
\int \csc ^{2} x e^{\cot x} d x
$$

Christy Galilei

Christy Galilei

Numerade Educator

Problem 33

Finding an Indefinite Integral In Exercises $15-46$ , find the indefinite integral.
$$
\int \frac{2}{e^{-x}+1} d x
$$

Pravakar Paul

Numerade Educator

Problem 34

Finding an Indefinite Integral In Exercises $15-46$ , find the indefinite integral.
$$
\int \frac{2}{7 e^{x}+4} d x
$$

Christy Galilei

Christy Galilei

Numerade Educator

Problem 35

Finding an Indefinite Integral In Exercises $15-46$ , find the indefinite integral.
$$
\int \frac{\ln x^{2}}{x} d x
$$

Pravakar Paul

Numerade Educator

Problem 36

Finding an Indefinite Integral In Exercises $15-46$ , find the indefinite integral.
$$
\int(\tan x)[\ln (\cos x)] d x
$$

Ashley Boni

Numerade Educator

Problem 37

Finding an Indefinite Integral In Exercises $15-46$ , find the indefinite integral.
$$
\int \frac{1+\cos \alpha}{\sin \alpha} d \alpha
$$

Pravakar Paul

Numerade Educator

Problem 38

Finding an Indefinite Integral In Exercises $15-46$ , find the indefinite integral.
$$
\int \frac{1}{\cos \theta-1} d \theta
$$

Ashley Boni

Numerade Educator

Problem 39

Finding an Indefinite Integral In Exercises $15-46$ , find the indefinite integral.
$$
\int \frac{-1}{\sqrt{1-(4 t+1)^{2}}} d t
$$

Pravakar Paul

Numerade Educator

Problem 40

Finding an Indefinite Integral In Exercises $15-46$ , find the indefinite integral.
$$
\int \frac{1}{25+4 x^{2}} d x
$$

Ashley Boni

Numerade Educator

Problem 41

Finding an Indefinite Integral In Exercises $15-46$ , find the indefinite integral.
$$
\int \frac{\tan (2 / t)}{t^{2}} d t
$$

CM

Caroline Morales

Numerade Educator

Problem 42

Finding an Indefinite Integral In Exercises $15-46$ , find the indefinite integral.
$$
\int \frac{e^{1 / t}}{t^{2}} d t
$$

Christy Galilei

Christy Galilei

Numerade Educator

Problem 43

Finding an Indefinite Integral In Exercises $15-46$ , find the indefinite integral.
$$
\int \frac{6}{\sqrt{10 x-x^{2}}} d x
$$

Dwijendra Rao

Numerade Educator

Problem 44

Finding an Indefinite Integral In Exercises $15-46$ , find the indefinite integral.
$$
\int \frac{1}{(x-1) \sqrt{4 x^{2}-8 x+3}} d x
$$

Ashley Boni

Numerade Educator

Problem 45

Finding an Indefinite Integral In Exercises $15-46$ , find the indefinite integral.
$$
\int \frac{4}{4 x^{2}+4 x+65} d x
$$

Pravakar Paul

Numerade Educator

Problem 46

Finding an Indefinite Integral In Exercises $15-46$ , find the indefinite integral.
$$
\int \frac{1}{x^{2}-4 x+9} d x
$$

Ashley Boni

Numerade Educator

Problem 47

Slope Field In Exercises 47 and 48 , a differential equation, a point, and a slope field are given. (a) Sketch two approximate solutions of the differential equation on the slope field, one of which passes through the given point. (b) Use integration to
find the particular solution of the differential equation and use a graphing utility to graph the solution. Compare the result with the sketches in part (a). To print an enlarged copy of the graph, go to MathGraphs.com.
$$
\begin{array}{l}{\frac{d s}{d t}=\frac{t}{\sqrt{1-t^{4}}}} \\ {\left(0,-\frac{1}{2}\right)}\end{array}
$$

Kian Manafi

Numerade Educator

Problem 48

Slope Field In Exercises 47 and 48 , a differential equation, a point, and a slope field are given. (a) Sketch two approximate solutions of the differential equation on the slope field, one of which passes through the given point. (b) Use integration to
find the particular solution of the differential equation and use a graphing utility to graph the solution. Compare the result with the sketches in part (a). To print an enlarged copy of the graph, go to MathGraphs.com.
$$
\begin{array}{l}{\frac{d y}{d x}=\frac{1}{\sqrt{4 x-x^{2}}}} \\ {\left(2, \frac{1}{2}\right)}\end{array}
$$

Carson Merrill

Numerade Educator

Problem 49

Slope Field In Exercises 49 and 50 , use a computer algebra system to graph the slope field for the differential equation and graph the solution through the specified initial condition.
$$
\frac{d y}{d x}=0.8 y, y(0)=4
$$

Dorcas Attuabea Addo

Dorcas Attuabea Addo

Numerade Educator

Problem 50

Slope Field In Exercises 49 and 50 , use a computer algebra system to graph the slope field for the differential equation and graph the solution through the specified initial condition.
$$
\frac{d y}{d x}=5-y, y(0)=1
$$

Dorcas Attuabea Addo

Dorcas Attuabea Addo

Numerade Educator

Problem 51

Differential Equation In Exercises $51-56,$ solve the differential equation.
$$
\frac{d y}{d x}=\left(e^{x}+5\right)^{2}
$$

Pravakar Paul

Numerade Educator

Problem 52

Differential Equation In Exercises $51-56,$ solve the differential equation.
$$
\frac{d y}{d x}=\left(4-e^{2 x}\right)^{2}
$$

Ashley Boni

Numerade Educator

Problem 53

Differential Equation In Exercises $51-56,$ solve the differential equation.
$$
\frac{d r}{d t}=\frac{10 e^{t}}{\sqrt{1-e^{2 t}}}
$$

Pravakar Paul

Numerade Educator

Problem 54

Differential Equation In Exercises $51-56,$ solve the differential equation.
$$
\frac{d r}{d t}=\frac{\left(1+e^{t}\right)^{2}}{e^{3 t}}
$$

Ashley Boni

Numerade Educator

Problem 55

Differential Equation In Exercises $51-56,$ solve the differential equation.
$$
\left(4+\tan ^{2} x\right) y^{\prime}=\sec ^{2} x
$$

Pravakar Paul

Numerade Educator

Problem 56

Differential Equation In Exercises $51-56,$ solve the differential equation.
$$
y^{\prime}=\frac{1}{x \sqrt{4 x^{2}-9}}
$$

Ashley Boni

Numerade Educator

Problem 57

Evaluating a Definite Integral In Exercises $57-64$ , evaluate the definite integral. Use the integration capabilities of a graphing utility to verify your result.
$$
\int_{0}^{\pi / 4} \cos 2 x d x
$$

Christy Galilei

Christy Galilei

Numerade Educator

Problem 58

Evaluating a Definite Integral In Exercises $57-64$ , evaluate the definite integral. Use the integration capabilities of a graphing utility to verify your result.
$$
\int_{0}^{\pi} \sin ^{2} t \cos t d t
$$

Ashley Boni

Numerade Educator

Problem 59

Evaluating a Definite Integral In Exercises $57-64$ , evaluate the definite integral. Use the integration capabilities of a graphing utility to verify your result.
$$
\int_{0}^{1} x e^{-x^{2}} d x
$$

Pravakar Paul

Numerade Educator

Problem 60

Evaluating a Definite Integral In Exercises $57-64$ , evaluate the definite integral. Use the integration capabilities of a graphing utility to verify your result.
$$
\int_{1}^{e} \frac{1-\ln x}{x} d x
$$

Christy Galilei

Christy Galilei

Numerade Educator

Problem 61

Evaluating a Definite Integral In Exercises $57-64$ , evaluate the definite integral. Use the integration capabilities of a graphing utility to verify your result.
$$
\int_{0}^{8} \frac{2 x}{\sqrt{x^{2}+36}} d x
$$

Pravakar Paul

Numerade Educator

Problem 62

Evaluating a Definite Integral In Exercises $57-64$ , evaluate the definite integral. Use the integration capabilities of a graphing utility to verify your result.
$$
\int_{1}^{3} \frac{2 x^{2}+3 x-2}{x} d x
$$

Ashley Boni

Numerade Educator

Problem 63

Evaluating a Definite Integral In Exercises $57-64$ , evaluate the definite integral. Use the integration capabilities of a graphing utility to verify your result.
$$
\int_{0}^{2 / \sqrt{3}} \frac{1}{4+9 x^{2}} d x
$$

Pravakar Paul

Numerade Educator

Problem 64

Evaluating a Definite Integral In Exercises $57-64$ , evaluate the definite integral. Use the integration capabilities of a graphing utility to verify your result.
$$
\int_{0}^{7} \frac{1}{\sqrt{100-x^{2}}} d x
$$

Ashley Boni

Numerade Educator

Problem 65

Area In Exercises $65-68,$ find the area of the region.
$$
y=(-4 x+6)^{3 / 2}
$$

Christy Galilei

Christy Galilei

Numerade Educator

Problem 66

Area In Exercises $65-68,$ find the area of the region.
$$
y=\frac{3 x+2}{x^{2}+9}
$$

Dwijendra Rao

Numerade Educator

Problem 67

Area In Exercises $65-68,$ find the area of the region.
$$
y^{2}=x^{2}\left(1-x^{2}\right)
$$

Dwijendra Rao

Numerade Educator

Problem 68

Area In Exercises $65-68,$ find the area of the region.
$$
y=\sin 2 x
$$

Dorcas Attuabea Addo

Dorcas Attuabea Addo

Numerade Educator

Problem 69

Finding an Integral Using Technology In Exercises $69-72$ , use a computer algebra system to find the integral. Use the computer algebra system to graph two antiderivatives. Describe the relationship between the graphs of the two anti derivatives.
$$
\int \frac{1}{x^{2}+4 x+13} d x
$$

Dorcas Attuabea Addo

Dorcas Attuabea Addo

Numerade Educator

Problem 70

Finding an Integral Using Technology In Exercises $69-72$ , use a computer algebra system to find the integral. Use the computer algebra system to graph two antiderivatives. Describe the relationship between the graphs of the two anti derivatives.
$$
\int \frac{x-2}{x^{2}+4 x+13} d x
$$

Dorcas Attuabea Addo

Dorcas Attuabea Addo

Numerade Educator

Problem 71

Finding an Integral Using Technology In Exercises $69-72$ , use a computer algebra system to find the integral. Use the computer algebra system to graph two antiderivatives. Describe the relationship between the graphs of the two anti derivatives.
$$
\int \frac{1}{1+\sin \theta} d \theta
$$

Dorcas Attuabea Addo

Dorcas Attuabea Addo

Numerade Educator

Problem 72

Finding an Integral Using Technology In Exercises $69-72$ , use a computer algebra system to find the integral. Use the computer algebra system to graph two antiderivatives. Describe the relationship between the graphs of the two anti derivatives.
$$
\int\left(\frac{e^{x}+e^{-x}}{2}\right)^{3} d x
$$

Dorcas Attuabea Addo

Dorcas Attuabea Addo

Numerade Educator

Problem 73

Choosing a Formula In Exercises $73-76,$ state the integration formula you would use to perform the integration. Explain why you chose that formula. Do not integrate.
$$
\int x\left(x^{2}+1\right)^{3} d x
$$

Dwijendra Rao

Numerade Educator

Problem 74

Choosing a Formula In Exercises $73-76,$ state the integration formula you would use to perform the integration. Explain why you chose that formula. Do not integrate.
$$
\int x \sec \left(x^{2}+1\right) \tan \left(x^{2}+1\right) d x
$$

Dwijendra Rao

Numerade Educator

Problem 75

Choosing a Formula In Exercises $73-76,$ state the integration formula you would use to perform the integration. Explain why you chose that formula. Do not integrate.
$$
\int \frac{x}{x^{2}+1} d x
$$

Dwijendra Rao

Numerade Educator

Problem 76

Choosing a Formula In Exercises $73-76,$ state the integration formula you would use to perform the integration. Explain why you chose that formula. Do not integrate.
$$
\int \frac{1}{x^{2}+1} d x
$$

Dwijendra Rao

Numerade Educator

Problem 77

Finding Constants Determine the constants $a$ and $b$ such that
$$
\sin x+\cos x=a \sin (x+b)
$$
Use this result to integrate
$$
\int \frac{d x}{\sin x+\cos x}
$$

Pravakar Paul

Numerade Educator

Problem 78

Deriving a Rule Show that
$$
\sec x=\frac{\sin x}{\cos x}+\frac{\cos x}{1+\sin x}
$$
Then use this identity to derive the basic integration rule
$$
\int \sec x d x=\ln |\sec x+\tan x|+C
$$

Dwijendra Rao

Numerade Educator

Problem 79

Area The graphs of $f(x)=x$ and $g(x)=a x^{2}$ intersect at the points $(0,0)$ and $(1 / a, 1 / a) .$ Find $a(a>0)$ such that the area of the region bounded by the graphs of these two functions is $\frac{2}{3}$ .

Ashley Boni

Numerade Educator

Problem 80

Think About It When evaluating
$$
\int_{-1}^{1} x^{2} d x
$$
is it appropriate to substitute
$$
u=x^{2}, \quad x=\sqrt{u}, \quad \text { and } \quad d x=\frac{d u}{2 \sqrt{u}}
$$
to obtain
$$
\frac{1}{2} \int_{1}^{1} \sqrt{u} d u=0 ?
$$

Darshan Maheshwari

Darshan Maheshwari

Numerade Educator

Problem 81

Comparing Antiderivatives
(a) Explain why the antiderivative $y_{1}=e^{x+C_{1}}$ is equivalent to the antiderivative $y_{2}=C e^{x} .$
(b) Explain why the antiderivative $y_{1}=\sec ^{2} x+C_{1}$ is equivalent to the antiderivative $y_{2}=\tan ^{2} x+C .$

Pravakar Paul

Numerade Educator

Problem 82

HOW DO YOU SEE IT? Using the graph, is $\int_{0}^{5} f(x) d x$ positive or negative? Explain.

Ashley Boni

Numerade Educator

Problem 83

Approximation In Exercises 83 and $84,$ determine which value best approximates the area of the region between the $x$ -axis and the function over the given interval. (Make your selection on the basis of a sketch of the region and not by integrating.)
$$
\begin{array}{l}{f(x)=\frac{4 x}{x^{2}+1}, \quad[0,2]} \\ {\left(\begin{array}{llll}{\text { (a) } 3} & {\text { (b) } 1} & {\text { (c) }-8 \quad \text { (d) } 8}\end{array}\right.}\end{array}
$$

Kian Manafi

Numerade Educator

Problem 84

Approximation In Exercises 83 and $84,$ determine which value best approximates the area of the region between the $x$ -axis and the function over the given interval. (Make your selection on the basis of a sketch of the region and not by integrating.)
$$
\begin{array}{ll}{f(x)=\frac{4}{x^{2}+1},} & {[0,2]} \\ {\begin{array}{llll}{\text { (a) } 3} & {\text { (b) } 1} & {\text { (c) }-4 \quad \text { (d) } 4} & {\text { (e) } 10}\end{array}}\end{array}
$$

Kian Manafi

Numerade Educator

Problem 85

Interpreting Integrals In Exercises 85 and $86,$ (a) sketch the region whose area is given by the integral, (b) sketch the solid whose volume is given by the integral when the disk method is used, and (c) sketch the solid whose volume is given
by the integral when the shell method is used. (There is more than one correct answer for each part.)
$$
\int_{0}^{2} 2 \pi x^{2} d x
$$

Carson Merrill

Numerade Educator

Problem 86

Interpreting Integrals In Exercises 85 and $86,$ (a) sketch the region whose area is given by the integral, (b) sketch the solid whose volume is given by the integral when the disk method is used, and (c) sketch the solid whose volume is given
by the integral when the shell method is used. (There is more than one correct answer for each part.)
$$
\int_{0}^{4} \pi y d y
$$

Carson Merrill

Numerade Educator

Problem 87

Volume The region bounded by $y=e^{-x^{2}}, y=0, x=0$ and $x=b(b>0)$ is revolved about the $y$ -axis.
(a) Find the volume of the solid generated when $b=1$
(b) Find $b$ such that the volume of the generated solid is $\frac{4}{3}$ cubic units.

Dorcas Attuabea Addo

Dorcas Attuabea Addo

Numerade Educator

Problem 88

Volume Consider the region bounded by the graphs of $x=0, y=\cos x^{2}, y=\sin x^{2},$ and $x=\sqrt{\pi} / 2 .$ Find the volume of the solid generated by revolving the region about the $y$ -axis.

Taylor Francis

Numerade Educator

Problem 89

Arc Length Find the arc length of the graph of $y=\ln (\sin x)$ from $x=\pi / 4$ to $x=\pi / 2$

Pravakar Paul

Numerade Educator

Problem 90

Arc Length Find the arc length of the graph of $y=\ln (\cos x)$ from $x=0$ to $x=\pi / 3$

Darshan Maheshwari

Darshan Maheshwari

Numerade Educator

Problem 91

Surface Area Find the area of the surface formed by revolving the graph of $y=2 \sqrt{x}$ on the interval $[0,9]$ about the $x$ -axis.

CN

Cindy Nguyen

Numerade Educator

Problem 92

Centroid Find the $x$ -coordinate of the centroid of the region bounded by the graphs of
$$
y=\frac{5}{\sqrt{25-x^{2}}}, \quad y=0, \quad x=0, \quad \text { and } \quad x=4
$$

Dwijendra Rao

Numerade Educator

Problem 93

Average Value of a Function In Exercises 93 and $94,$ find the average value of the function over the given interval.
$$
f(x)=\frac{1}{1+x^{2}}, \quad-3 \leq x \leq 3
$$

Dwijendra Rao

Numerade Educator

Problem 94

Average Value of a Function In Exercises 93 and $94,$ find the average value of the function over the given interval.
$f(x)=\sin n x, \quad 0 \leq x \leq \pi / n, n$ is a positive integer.

Dwijendra Rao

Numerade Educator

Problem 95

Arc Length In Exercises 95 and $96,$ use the integration capabilities of a graphing utility to approximate the are length of the curve over the given interval.
$$
y=\tan \pi x, \quad\left[0, \frac{1}{4}\right]
$$

Kian Manafi

Numerade Educator

Problem 96

Arc Length In Exercises 95 and $96,$ use the integration capabilities of a graphing utility to approximate the are length of the curve over the given interval.
$$
y=x^{2 / 3}, \quad[1,8]
$$

Kian Manafi

Numerade Educator

Problem 97

Finding a Pattern
(a) Find $\int \cos ^{3} x d x .$
(b) Find $\int \cos ^{5} x d x$
(c) Find $\int \cos ^{7} x d x$
(d) Explain how to find $\int \cos ^{15} x d x$ without actually integrating.

Pravakar Paul

Numerade Educator

Problem 98

Finding a Pattern
(a) Write $\int \tan ^{3} x d x$ in terms of $\int \tan x d x$ . Then find $\int \tan ^{3} x d x .$
(b) Write $\int \tan ^{5} x d x$ in terms of $\int \tan ^{3} x d x$
(c) Write $\int \tan ^{2 k+1} x d x,$ where $k$ is a positive integer, in terms of $\int \tan ^{2 k-1} x d x$ .
(d) Explain how to find $\int \tan ^{15} x d x$ without actually integrating.

Dorcas Attuabea Addo

Dorcas Attuabea Addo

Numerade Educator

Problem 99

Methods of Integration Show that the following results are equivalent. Integration by tables:
$$
\int \sqrt{x^{2}+1} d x=\frac{1}{2}\left(x \sqrt{x^{2}+1}+\ln \left|x+\sqrt{x^{2}+1}\right|\right)+C
$$
Integration by computer algebra system:
$$
\int \sqrt{x^{2}+1} d x=\frac{1}{2}\left[x \sqrt{x^{2}+1}+\operatorname{arcsinh}(x)\right]+C
$$

Dorcas Attuabea Addo

Dorcas Attuabea Addo

Numerade Educator

Problem 100

Evaluate $\int_{2}^{4} \frac{\sqrt{\ln (9-x)} d x}{\sqrt{\ln (9-x)}+\sqrt{\ln (x+3)}}$

Darshan Maheshwari

Darshan Maheshwari

Numerade Educator