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Calculus of a Single Variable

Ron Larson, Bruce Edwards

Chapter 8

Integration Techniques, L’Hopital’s Rule, and Improper Integrals

Educators


Problem 1

Select the correct anti derivative.
$$\frac{d y}{d x}=\frac{x}{\sqrt{x^{2}+1}}$$
$$\begin{array}{ll}{\text { (a) } 2 \sqrt{x^{2}+1}+C} & {\text { (b) } \sqrt{x^{2}+1}+C} \\ {\text { (c) } \frac{1}{2} \sqrt{x^{2}+1}+C} & {\text { (d) } \ln \left(x^{2}+1\right)+C}\end{array}$$

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Problem 2

Select the correct anti derivative.
$$\frac{d y}{d x}=\frac{x}{x^{2}+1}$$
$$\begin{array}{ll}{\text { (a) } \ln \sqrt{x^{2}+1}+C} & {\text { (b) } \frac{2 x}{\left(x^{2}+1\right)^{2}}+C} \\ {\text { (c) } \arctan x+C} & {\text { (d) } \ln \left(x^{2}+1\right)+C}\end{array}$$

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Problem 3

Select the correct anti derivative.
$$\frac{d y}{d x}=\frac{1}{x^{2}+1}$$
$$\begin{array}{ll}{\text { (a) } \ln \sqrt{x^{2}+1}+C} & {\text { (b) } \frac{2 x}{\left(x^{2}+1\right)^{2}}+C} \\ {\text { (c) } \arctan x+C} & {\text { (d) } \ln \left(x^{2}+1\right)+C}\end{array}$$

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Problem 4

Select the correct anti derivative.
$$\frac{d y}{d x}=x \cos \left(x^{2}+1\right)$$
$$\begin{array}{ll}{\text { (a) } 2 x \sin \left(x^{2}+1\right)+C} & {\text { (b) }-\frac{1}{2} \sin \left(x^{2}+1\right)+C} \\ {\text { (c) } \frac{1}{2} \sin \left(x^{2}+1\right)+C} & {\text { (d) }-2 x \sin \left(x^{2}+1\right)+C}\end{array}$$

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Problem 5

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

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Problem 6

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

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Problem 7

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

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Problem 8

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

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Problem 9

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

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Problem 10

Select the basic integration formula you can use to find the integral, and identify $u$ and $a$ when appropriate.
$$\int \frac{-2 x}{\sqrt{x^{2}-4}} d x$$

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Problem 11

Select the basic integration formula you can use to find the integral, and identify $u$ and $a$ when appropriate.
$$\int t \sin t^{2} d t$$

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Problem 12

Select the basic integration formula you can use to find the integral, and identify $u$ and $a$ when appropriate.
$$\int \sec 5 x \tan 5 x d x$$

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Problem 13

Select the basic integration formula you can use to find the integral, and identify $u$ and $a$ when appropriate.
$$\int(\cos x) e^{\sin x} d x$$

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Problem 14

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

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Problem 15

Find the indefinite integral.
$$\int 14(x-5)^{6} d x$$

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Problem 16

Find the indefinite integral.
$$\int \frac{9}{(t-8)^{2}} d t$$

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Problem 17

Find the indefinite integral.
$$\int \frac{7}{(z-10)^{7}} d z$$

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Problem 18

Find the indefinite integral.
$$\int t^{2} \sqrt[3]{t^{3}-1} d t$$

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Problem 19

Find the indefinite integral.
$$\int\left[v+\frac{1}{(3 v-1)^{3}}\right] d v$$

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Problem 20

Find the indefinite integral.
$$\int\left[x-\frac{5}{(3 x+5)^{2}}\right] d x$$

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Problem 21

Find the indefinite integral.
$$\int \frac{t^{2}-3}{-t^{3}+9 t+1} d t$$

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Problem 22

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

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Problem 23

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

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Problem 24

Find the indefinite integral.
$$\int \frac{4 x}{x-8} d x$$

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Problem 25

Find the indefinite integral.
$$\int \frac{e^{x}}{1+e^{x}} d x$$

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Problem 26

Find the indefinite integral.
$$\int\left(\frac{1}{7 x-2}-\frac{1}{7 x+2}\right) d x$$

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Problem 27

Find the indefinite integral.
$$\int\left(5+4 x^{2}\right)^{2} d x$$

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Problem 28

Find the indefinite integral.
$$\int x\left(1+\frac{1}{x}\right)^{3} d x$$

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Problem 29

Find the indefinite integral.
$$\int x \cos 2 \pi x^{2} d x$$

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Problem 30

Find the indefinite integral.
$$\int \sec 4 x d x$$

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Problem 31

Find the indefinite integral.
$$\int \csc \pi x \cot \pi x d x$$

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Problem 32

Find the indefinite integral.
$$\int \frac{\sin x}{\sqrt{\cos x}} d x$$

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Problem 33

Find the indefinite integral.
$$\int e^{11 x} d x$$

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Problem 34

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

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Problem 35

Find the indefinite integral.
$$\int \frac{2}{e^{-x}+1} d x$$

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Problem 36

Find the indefinite integral.
$$\int \frac{5}{3 e^{x}-2} d x$$

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Problem 37

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

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Problem 38

Find the indefinite integral.
$$\int(\tan x)[\ln (\cos x)] d x$$

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Problem 39

Find the indefinite integral.
$$\int \frac{1+\sin x}{\cos x} d x$$

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Problem 40

Find the indefinite integral.
$$\int \frac{1+\cos \alpha}{\sin \alpha} d \alpha$$

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Problem 41

Find the indefinite integral.
$$\int \frac{1}{\cos \theta-1} d \theta$$

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Problem 42

Find the indefinite integral.
$$\int \frac{2}{3(\sec x-1)} d x$$

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Problem 43

Find the indefinite integral.
$$\int \frac{-1}{\sqrt{1-(4 t+1)^{2}}} d t$$

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Problem 44

Find the indefinite integral.
$$\int \frac{1}{9+5 x^{2}} d x$$

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Problem 45

Find the indefinite integral.
$$\int \frac{\tan (2 / t)}{t^{2}} d t$$

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Problem 46

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

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Problem 47

Find the indefinite integral.
$$\int \frac{6}{\sqrt{10 x-x^{2}}} d x$$

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Problem 48

Find the indefinite integral.
$$\int \frac{1}{(x-1) \sqrt{4 x^{2}-8 x+3}} d x$$

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Problem 49

Find the indefinite integral.
$$\int \frac{4}{4 x^{2}+4 x+65} d x$$

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Problem 50

Find the indefinite integral.
$$\int \frac{1}{x^{2}-4 x+9} d x$$

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Problem 51

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

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Problem 52

Find the indefinite integral.
$$\int \frac{12}{\sqrt{3-8 x-x^{2}}} d x$$

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Problem 53

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 the website www.mathgraphs.com.
$$\begin{array}{l}{\frac{d s}{d t}=\frac{t}{\sqrt{1-t^{4}}}} \\ {\left(0,-\frac{1}{2}\right)}\end{array}$$

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Problem 54

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 the website www.mathgraphs.com.
$$\begin{array}{l}{\frac{d y}{d x}=\tan ^{2}(2 x)} \\ {(0,0)}\end{array}$$

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Problem 55

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 the website www.mathgraphs.com.
$$\begin{array}{l}{\frac{d y}{d x}=(\sec x+\tan x)^{2}} \\ {(0,1)}\end{array}$$

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Problem 56

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 the website www.mathgraphs.com.
$$\begin{array}{l}{\frac{d y}{d x}=\frac{1}{\sqrt{4 x-x^{2}}}} \\ {\left(2, \frac{1}{2}\right)}\end{array}$$

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Problem 57

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

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Problem 58

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

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Problem 59

Solve the differential equation.
$$\frac{d y}{d x}=\left(e^{x}+5\right)^{2}$$

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Problem 60

Solve the differential equation.
$$\frac{d y}{d x}=\left(3-e^{x}\right)^{2}$$

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Problem 61

Solve the differential equation.
$$\frac{d r}{d t}=\frac{10 e^{t}}{\sqrt{1-e^{2 t}}}$$

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Problem 62

Solve the differential equation.
$$\frac{d r}{d t}=\frac{\left(1+e^{t}\right)^{2}}{e^{t}}$$

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Problem 63

Solve the differential equation.
$$\left(4+\tan ^{2} x\right) y^{\prime}=\sec ^{2} x$$

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Problem 64

Solve the differential equation.
$$y^{\prime}=\frac{1}{x \sqrt{4 x^{2}-1}}$$

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Problem 65

Evaluate the definite integral. Use the integration capabilities of a graphing utility to verify your result.
$$\int_{0}^{\pi / 4} \cos 2 x d x$$

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Problem 66

Evaluate the definite integral. Use the integration capabilities of a graphing utility to verify your result.
$$\int_{0}^{\pi} \sin ^{2} t \cos t d t$$

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Problem 67

Evaluate the definite integral. Use the integration capabilities of a graphing utility to verify your result.
$$\int_{0}^{1} x e^{-x^{2}} d x$$

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Problem 68

Evaluate the definite integral. Use the integration capabilities of a graphing utility to verify your result.
$$\int_{1}^{e} \frac{1-\ln x}{x} d x$$

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Problem 69

Evaluate the definite integral. Use the integration capabilities of a graphing utility to verify your result.
$$\int_{0}^{8} \frac{2 x}{\sqrt{x^{2}+36}} d x$$

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Problem 70

Evaluate the definite integral. Use the integration capabilities of a graphing utility to verify your result.
$$\int_{1}^{2} \frac{x-2}{x} d x$$

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Problem 71

Evaluate the definite integral. Use the integration capabilities of a graphing utility to verify your result.
$$\int_{0}^{2 / \sqrt{3}} \frac{1}{4+9 x^{2}} d x$$

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Problem 72

Evaluate the definite integral. Use the integration capabilities of a graphing utility to verify your result.
$$\int_{0}^{7} \frac{1}{\sqrt{100-x^{2}}} d x$$

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Problem 73

Find the area of the region.
$$y=(-4 x+6)^{3 / 2}$$

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Problem 74

Find the area of the region.
$$y=x \sqrt{8-2 x^{2}}$$

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Problem 75

Find the area of the region.
$$y=\frac{3 x+2}{x^{2}+9}$$

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Problem 76

Find the area of the region.
$$y=\frac{5}{x^{2}+1}$$

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Problem 77

Find the area of the region.
$$y^{2}=x^{2}\left(1-x^{2}\right)$$

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Problem 78

Find the area of the region.
$$y=\sin 2 x$$

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Problem 79

Use a computer algebra system to find the integral. Use the computer algebra system to graph two anti derivatives. Describe the relationship between the graphs of the two anti derivatives.
$$\int \frac{1}{x^{2}+4 x+13} d x$$

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Problem 80

Use a computer algebra system to find the integral. Use the computer algebra system to graph two anti derivatives. Describe the relationship between the graphs of the two anti derivatives.
$$\int \frac{x-2}{x^{2}+4 x+13} d x$$

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Problem 81

Use a computer algebra system to find the integral. Use the computer algebra system to graph two anti derivatives. Describe the relationship between the graphs of the two anti derivatives.
$$\int \frac{1}{1+\sin \theta} d \theta$$

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Problem 82

Use a computer algebra system to find the integral. Use the computer algebra system to graph two anti derivatives. Describe the relationship between the graphs of the two anti derivatives.
$$\int\left(\frac{e^{x}+e^{-x}}{2}\right)^{3} d x$$

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Problem 83

State the integration formula you would use to perform the integration. Explain why you chose that formula. Do not integrate.
$$\int x\left(x^{2}+1\right)^{3} d x$$

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Problem 84

State the integration formula you would use to perform the integration. Explain why you chose that formula. Do not integrate.
$$\int x \sec \left(x^{2}+1\right) \tan \left(x^{2}+1\right) d x$$

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Problem 85

State the integration formula you would use to perform the integration. Explain why you chose that formula. Do not integrate.
$$\int \frac{x}{x^{2}+1} d x$$

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Problem 86

State the integration formula you would use to perform the integration. Explain why you chose that formula. Do not integrate.
$$\int \frac{1}{x^{2}+1} d x$$

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Problem 87

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

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Problem 88

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
$\int \sec x d x=\ln |\sec x+\tan x|+C$.

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Problem 89

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 $\frac{2}{3}$ .

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Problem 90

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

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Problem 91

Use a graphing utility to graph the function $f(x)=\frac{1}{5}\left(x^{3}-7 x^{2}+10 x\right) .$ Use the graph to determine whether $\int_{0}^{5} f(x) d x$ is positive or negative. Explain.

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Problem 92

When evaluating $\int_{-1}^{1} x^{2} d x,$ is it appropriate to substitute $u=x^{2}, x=\sqrt{u},$ and $d x=\frac{d u}{2 \sqrt{u}}$ to obtain $\frac{1}{2} \int_{1}^{1} \sqrt{u} d u=0 ?$ Explain.

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Problem 93

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.)
$$f(x)=\frac{4 x}{x^{2}+1}, \quad[0,2]$$
$$\begin{array}{llllll}{\text { (a) } 3} & {\text { (b) } 1} & {\text { (c) }-8} & {\text { (d) } 8} & {\text { (e) } 10}\end{array}$$

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Problem 94

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.)
$$f(x)=\frac{4}{x^{2}+1}, \quad[0,2]$$
$$\begin{array}{llllll}{\text { (a) } 3} & {\text { (b) } 1} & {\text { (c) }-4} & {\text { (d) } 4} & {\text { (e) } 10}\end{array}$$

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Problem 95

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

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Problem 96

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

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Problem 97

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 if $b=1$
(b) Find $b$ such that the volume of the generated solid is $\frac{4}{3}$ cubic units.

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Problem 98

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.

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Problem 99

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

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Problem 100

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

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Problem 101

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.

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Problem 102

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

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Problem 103

Find the average value of the function over the given interval.
$$f(x)=\frac{1}{1+x^{2}}, \quad-3 \leq x \leq 3$$

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Problem 104

Find the average value of the function over the given interval.
$$\begin{array}{l}{f(x)=\sin n x, \quad 0 \leq x \leq \pi / n, n \text { is a positive integer. }}\end{array}$$

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Problem 105

Use the integration capabilities of a graphing utility to approximate the are length of the curve over the given interval.
$$y=\tan \pi x, \quad\left[0, \frac{1}{4}\right]$$

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Problem 106

Use the integration capabilities of a graphing utility to approximate the are length of the curve over the given interval.
$$y=x^{2 / 3}, \quad[1,8]$$

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Problem 107

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.

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Problem 108

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.

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Problem 109

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:
$\int \sqrt{x^{2}+1} d x=\frac{1}{2}\left[x \sqrt{x^{2}+1}+\operatorname{arcsinh}(x)\right]+C$

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Problem 110

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

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