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  1. Preface
  2. 1 Integration
    1. Introduction
    2. 1.1 Approximating Areas
    3. 1.2 The Definite Integral
    4. 1.3 The Fundamental Theorem of Calculus
    5. 1.4 Integration Formulas and the Net Change Theorem
    6. 1.5 Substitution
    7. 1.6 Integrals Involving Exponential and Logarithmic Functions
    8. 1.7 Integrals Resulting in Inverse Trigonometric Functions
    9. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
      4. Review Exercises
  3. 2 Applications of Integration
    1. Introduction
    2. 2.1 Areas between Curves
    3. 2.2 Determining Volumes by Slicing
    4. 2.3 Volumes of Revolution: Cylindrical Shells
    5. 2.4 Arc Length of a Curve and Surface Area
    6. 2.5 Physical Applications
    7. 2.6 Moments and Centers of Mass
    8. 2.7 Integrals, Exponential Functions, and Logarithms
    9. 2.8 Exponential Growth and Decay
    10. 2.9 Calculus of the Hyperbolic Functions
    11. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
      4. Review Exercises
  4. 3 Techniques of Integration
    1. Introduction
    2. 3.1 Integration by Parts
    3. 3.2 Trigonometric Integrals
    4. 3.3 Trigonometric Substitution
    5. 3.4 Partial Fractions
    6. 3.5 Other Strategies for Integration
    7. 3.6 Numerical Integration
    8. 3.7 Improper Integrals
    9. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
      4. Review Exercises
  5. 4 Introduction to Differential Equations
    1. Introduction
    2. 4.1 Basics of Differential Equations
    3. 4.2 Direction Fields and Numerical Methods
    4. 4.3 Separable Equations
    5. 4.4 The Logistic Equation
    6. 4.5 First-order Linear Equations
    7. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
      4. Review Exercises
  6. 5 Sequences and Series
    1. Introduction
    2. 5.1 Sequences
    3. 5.2 Infinite Series
    4. 5.3 The Divergence and Integral Tests
    5. 5.4 Comparison Tests
    6. 5.5 Alternating Series
    7. 5.6 Ratio and Root Tests
    8. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
      4. Review Exercises
  7. 6 Power Series
    1. Introduction
    2. 6.1 Power Series and Functions
    3. 6.2 Properties of Power Series
    4. 6.3 Taylor and Maclaurin Series
    5. 6.4 Working with Taylor Series
    6. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
      4. Review Exercises
  8. 7 Parametric Equations and Polar Coordinates
    1. Introduction
    2. 7.1 Parametric Equations
    3. 7.2 Calculus of Parametric Curves
    4. 7.3 Polar Coordinates
    5. 7.4 Area and Arc Length in Polar Coordinates
    6. 7.5 Conic Sections
    7. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
      4. Review Exercises
  9. A | Table of Integrals
  10. B | Table of Derivatives
  11. C | Review of Pre-Calculus
  12. Answer Key
    1. Chapter 1
    2. Chapter 2
    3. Chapter 3
    4. Chapter 4
    5. Chapter 5
    6. Chapter 6
    7. Chapter 7
  13. Index

Checkpoint

3.1

x e 2 x d x = 1 2 x e 2 x 1 4 e 2 x + C x e 2 x d x = 1 2 x e 2 x 1 4 e 2 x + C

3.2

1 2 x 2 ln x 1 4 x 2 + C 1 2 x 2 ln x 1 4 x 2 + C

3.3

x 2 cos x + 2 x sin x + 2 cos x + C x 2 cos x + 2 x sin x + 2 cos x + C

3.4

π 2 1 π 2 1

3.5

1 5 sin 5 x + C 1 5 sin 5 x + C

3.6

1 3 sin 3 x 1 5 sin 5 x + C 1 3 sin 3 x 1 5 sin 5 x + C

3.7

1 2 x + 1 4 sin ( 2 x ) + C 1 2 x + 1 4 sin ( 2 x ) + C

3.8

sin x 1 3 sin 3 x + C sin x 1 3 sin 3 x + C

3.9

1 2 x + 1 12 sin ( 6 x ) + C 1 2 x + 1 12 sin ( 6 x ) + C

3.10

1 2 sin x + 1 22 sin ( 11 x ) + C 1 2 sin x + 1 22 sin ( 11 x ) + C

3.11

1 6 tan 6 x + C 1 6 tan 6 x + C

3.12

1 9 sec 9 x 1 7 sec 7 x + C 1 9 sec 9 x 1 7 sec 7 x + C

3.13

sec 5 x d x = 1 4 sec 3 x tan x + 3 4 sec 3 x sec 5 x d x = 1 4 sec 3 x tan x + 3 4 sec 3 x

3.14

125 sin 3 θ d θ 125 sin 3 θ d θ

3.15

32 tan 3 θ sec 3 θ d θ 32 tan 3 θ sec 3 θ d θ

3.16

ln | x 2 + x 2 4 2 | + C ln | x 2 + x 2 4 2 | + C

3.17

x 5 ln | x + 2 | + C x 5 ln | x + 2 | + C

3.18

2 5 ln | x + 3 | + 3 5 ln | x 2 | + C 2 5 ln | x + 3 | + 3 5 ln | x 2 | + C

3.19

x + 2 ( x + 3 ) 3 ( x 4 ) 2 = A x + 3 + B ( x + 3 ) 2 + C ( x + 3 ) 3 + D ( x 4 ) + E ( x 4 ) 2 x + 2 ( x + 3 ) 3 ( x 4 ) 2 = A x + 3 + B ( x + 3 ) 2 + C ( x + 3 ) 3 + D ( x 4 ) + E ( x 4 ) 2

3.20

x 2 + 3 x + 1 ( x + 2 ) ( x 3 ) 2 ( x 2 + 4 ) 2 = A x + 2 + B x 3 + C ( x 3 ) 2 + D x + E x 2 + 4 + F x + G ( x 2 + 4 ) 2 x 2 + 3 x + 1 ( x + 2 ) ( x 3 ) 2 ( x 2 + 4 ) 2 = A x + 2 + B x 3 + C ( x 3 ) 2 + D x + E x 2 + 4 + F x + G ( x 2 + 4 ) 2

3.21

Possible solutions include sinh−1(x2)+Csinh−1(x2)+C and ln|x2+4+x|+C.ln|x2+4+x|+C.

3.22

24 35 24 35

3.23

17 24 17 24

3.24

0.0074, 1.1%

3.25

1 192 1 192

3.26

25 36 25 36

3.27

e3,e3, converges

3.28

+,+, diverges

3.29

Since e+1xdx=+,e+1xdx=+, e+lnxxdxe+lnxxdx diverges.

Section 3.1 Exercises

1 .

u = x 3 u = x 3

3 .

u = y 3 u = y 3

5 .

u = sin ( 2 x ) u = sin ( 2 x )

7 .

x + x ln x + C x + x ln x + C

9 .

x tan −1 x 1 2 ln ( 1 + x 2 ) + C x tan −1 x 1 2 ln ( 1 + x 2 ) + C

11 .

1 2 x cos ( 2 x ) + 1 4 sin ( 2 x ) + C 1 2 x cos ( 2 x ) + 1 4 sin ( 2 x ) + C

13 .

e x ( −1 x ) + C e x ( −1 x ) + C

15 .

2 x cos x + ( −2 + x 2 ) sin x + C 2 x cos x + ( −2 + x 2 ) sin x + C

17 .

1 2 ( 1 + 2 x ) ( −1 + ln ( 1 + 2 x ) ) + C 1 2 ( 1 + 2 x ) ( −1 + ln ( 1 + 2 x ) ) + C

19 .

1 2 e x ( cos x + sin x ) + C 1 2 e x ( cos x + sin x ) + C

21 .

e x 2 2 + C e x 2 2 + C

23 .

1 2 x cos [ ln ( 2 x ) ] + 1 2 x sin [ ln ( 2 x ) ] + C 1 2 x cos [ ln ( 2 x ) ] + 1 2 x sin [ ln ( 2 x ) ] + C

25 .

2 x 2 x ln x + x ( ln x ) 2 + C 2 x 2 x ln x + x ( ln x ) 2 + C

27 .

( x 3 9 + 1 3 x 3 ln x ) + C ( x 3 9 + 1 3 x 3 ln x ) + C

29 .

1 2 1 4 x 2 + x cos −1 ( 2 x ) + C 1 2 1 4 x 2 + x cos −1 ( 2 x ) + C

31 .

( −2 + x 2 ) cos x + 2 x sin x + C ( −2 + x 2 ) cos x + 2 x sin x + C

33 .

x ( −6 + x 2 ) cos x + 3 ( −2 + x 2 ) sin x + C x ( −6 + x 2 ) cos x + 3 ( −2 + x 2 ) sin x + C

35 .

1 2 x ( 1 1 x 2 + x · sec −1 x ) + C 1 2 x ( 1 1 x 2 + x · sec −1 x ) + C

37 .

cosh x + x sinh x + C cosh x + x sinh x + C

39 .

1 4 3 4 e 2 1 4 3 4 e 2

41 .

2

43 .

2 π 2 π

45 .

−2 + π −2 + π

47 .

sin ( x ) + ln [ sin ( x ) ] sin x + C sin ( x ) + ln [ sin ( x ) ] sin x + C

49 .

Answers vary

51 .

a. 25(1+x)(−3+2x)3/2+C25(1+x)(−3+2x)3/2+C b. 25(1+x)(−3+2x)3/2+C25(1+x)(−3+2x)3/2+C

53 .

Do not use integration by parts. Choose u to be lnx,lnx, and the integral is of the form u2du.u2du.

55 .

Do not use integration by parts. Let u=x23,u=x23, and the integral can be put into the form eudu.eudu.

57 .

Do not use integration by parts. Choose u to be u=3x3+2u=3x3+2 and the integral can be put into the form sin(u)du.sin(u)du.

59 .

The area under graph is 0.39535.

This figure is the graph of y=e^-x sin(pi*x). The curve begins in the third quadrant at x=0.5, increases through the origin, reaches a high point between 0.5 and 0.75, then decreases, passing through x=1.
61 .

2 π e 2 π e

63 .

2.05

65 .

12 π 12 π

67 .

8 π 2 8 π 2

Section 3.2 Exercises

69 .

cos 2 x cos 2 x

71 .

1 cos ( 2 x ) 2 1 cos ( 2 x ) 2

73 .

sin 4 x 4 + C sin 4 x 4 + C

75 .

1 12 tan 6 ( 2 x ) + C 1 12 tan 6 ( 2 x ) + C

77 .

sec 2 ( x 2 ) + C sec 2 ( x 2 ) + C

79 .

cos x + 1 3 cos 2 x + C cos x + 1 3 cos 2 x + C

81 .

12cos2x+C12cos2x+C or 12sin2x+C12sin2x+C

83 .

1 3 cos 3 x + 2 5 cos 5 x 1 7 cos 7 x + C 1 3 cos 3 x + 2 5 cos 5 x 1 7 cos 7 x + C

85 .

2 3 ( sin x ) 3 2 + C 2 3 ( sin x ) 3 2 + C

87 .

sec x + C sec x + C

89 .

1 2 sec x tan x 1 2 ln ( sec x + tan x ) + C 1 2 sec x tan x 1 2 ln ( sec x + tan x ) + C

91 .

2tanx3+13sec(x)2tanx2tanx3+13sec(x)2tanx =tanx+tan3x3+C=tanx+tan3x3+C

93 .

ln | cot x + csc x | + C ln | cot x + csc x | + C

95 .

sin 3 ( a x ) 3 a + C sin 3 ( a x ) 3 a + C

97 .

π 2 π 2

99 .

x 2 + 1 12 sin ( 6 x ) + C x 2 + 1 12 sin ( 6 x ) + C

101 .

x+Cx+C

103 .

0

105 .

0

107 .

0

109 .

Approximately 0.239

111 .

2 2

113 .

1.0

115 .

0

117 .

3 θ 8 1 4 π sin ( 2 π θ ) + 1 32 π sin ( 4 π θ ) + C = f ( x ) 3 θ 8 1 4 π sin ( 2 π θ ) + 1 32 π sin ( 4 π θ ) + C = f ( x )

119 .

ln ( 3 ) ln ( 3 )

121 .

π π sin ( 2 x ) cos ( 3 x ) d x = 0 π π sin ( 2 x ) cos ( 3 x ) d x = 0

123 .

tan ( x ) x ( 8 tan x 21 + 2 7 sec x 2 tan x ) + C = f ( x ) tan ( x ) x ( 8 tan x 21 + 2 7 sec x 2 tan x ) + C = f ( x )

125 .

The second integral is more difficult because the first integral is simply a u-substitution type.

Section 3.3 Exercises

127 .

9 tan 2 θ 9 tan 2 θ

129 .

a 2 cosh 2 θ a 2 cosh 2 θ

131 .

4 ( x 1 2 ) 2 4 ( x 1 2 ) 2

133 .

( x + 1 ) 2 + 5 ( x + 1 ) 2 + 5

135 .

ln | x + a 2 + x 2 | + C ln | x + a 2 + x 2 | + C

137 .

1 3 ln | 9 x 2 + 1 + 3 x | + C 1 3 ln | 9 x 2 + 1 + 3 x | + C

139 .

1 x 2 x + C 1 x 2 x + C

141 .

9 [ x x 2 + 9 18 + 1 2 l n | x 2 + 9 3 + x 3 | ] + C 9 [ x x 2 + 9 18 + 1 2 l n | x 2 + 9 3 + x 3 | ] + C

143 .

1 3 9 θ 2 ( 18 + θ 2 ) + C 1 3 9 θ 2 ( 18 + θ 2 ) + C

145 .

( −1 + x 2 ) ( 2 + 3 x 2 ) x 6 x 8 15 x 3 + C ( −1 + x 2 ) ( 2 + 3 x 2 ) x 6 x 8 15 x 3 + C

147 .

x 9 −9 + x 2 + C x 9 −9 + x 2 + C

149 .

1 2 ( ln | x + x 2 1 | + x x 2 1 ) + C 1 2 ( ln | x + x 2 1 | + x x 2 1 ) + C

151 .

1 + x 2 x + C 1 + x 2 x + C

153 .

1 8 ( x ( 5 2 x 2 ) 1 x 2 + 3 arcsin x ) + C 1 8 ( x ( 5 2 x 2 ) 1 x 2 + 3 arcsin x ) + C

155 .

ln x ln | 1 + 1 x 2 | + C ln x ln | 1 + 1 x 2 | + C

157 .

−1 + x 2 x + ln | x + −1 + x 2 | + C −1 + x 2 x + ln | x + −1 + x 2 | + C

159 .

1 + x 2 x + arcsinh x + C 1 + x 2 x + arcsinh x + C

161 .

1 1 + x + C 1 1 + x + C

163 .

2 −10 + x x ln | −10 + x + x | ( 10 x ) x + C 2 −10 + x x ln | −10 + x + x | ( 10 x ) x + C

165 .

9π2;9π2; area of a semicircle with radius 3

167 .

arcsin(x)+Carcsin(x)+C is the common answer.

169 .

12ln(1+x2)+C12ln(1+x2)+C is the result using either method.

171 .

Use trigonometric substitution. Let x=sec(θ).x=sec(θ).

173 .

4.367

175 .

π 2 8 + π 4 π 2 8 + π 4

177 .

y = 1 16 ln | x + 8 x 8 | + 3 y = 1 16 ln | x + 8 x 8 | + 3

179 .

24.6 m3

181 .

2 π 3 2 π 3

Section 3.4 Exercises

183 .

2 x + 1 + 5 2 ( x + 2 ) + 1 2 x 2 x + 1 + 5 2 ( x + 2 ) + 1 2 x

185 .

1 x 2 + 3 x 1 x 2 + 3 x

187 .

2 x 2 + 4 x + 8 + 16 x 2 2 x 2 + 4 x + 8 + 16 x 2

189 .

1 x 2 1 x + 1 x 1 1 x 2 1 x + 1 x 1

191 .

1 2 ( x 2 ) + 1 2 ( x 1 ) 1 6 x + 1 6 ( x 3 ) 1 2 ( x 2 ) + 1 2 ( x 1 ) 1 6 x + 1 6 ( x 3 )

193 .

1 x 1 + 2 x + 1 x 2 + x + 1 1 x 1 + 2 x + 1 x 2 + x + 1

195 .

2 x + 1 + x x 2 + 4 1 ( x 2 + 4 ) 2 2 x + 1 + x x 2 + 4 1 ( x 2 + 4 ) 2

197 .

ln | 2 x | + 2 ln | 4 + x | + C ln | 2 x | + 2 ln | 4 + x | + C

199 .

1 2 ln | 4 x 2 | + C 1 2 ln | 4 x 2 | + C

201 .

2 ( x + 1 3 arctan ( 1 + x 3 ) ) + C 2 ( x + 1 3 arctan ( 1 + x 3 ) ) + C

203 .

2 ln | x | 3 ln | 1 + x | + C 2 ln | x | 3 ln | 1 + x | + C

205 .

1 16 ( 4 −2 + x ln | −2 + x | + ln | 2 + x | ) + C 1 16 ( 4 −2 + x ln | −2 + x | + ln | 2 + x | ) + C

207 .

1 30 ( −2 5 arctan [ 1 + x 5 ] + 2 ln | −4 + x | ln | 6 + 2 x + x 2 | ) + C 1 30 ( −2 5 arctan [ 1 + x 5 ] + 2 ln | −4 + x | ln | 6 + 2 x + x 2 | ) + C

209 .

3 x + 4 ln | x + 2 | + x + C 3 x + 4 ln | x + 2 | + x + C

211 .

ln | 3 x | + 1 2 ln | x 2 + 4 | + C ln | 3 x | + 1 2 ln | x 2 + 4 | + C

213 .

ln | x 2 | 1 2 ln | x 2 + 2 x + 2 | + C ln | x 2 | 1 2 ln | x 2 + 2 x + 2 | + C

215 .

x + ln | 1 e x | + C x + ln | 1 e x | + C

217 .

1 5 ln | cos x + 3 cos x 2 | + C 1 5 ln | cos x + 3 cos x 2 | + C

219 .

1 2 2 e 2 t + C 1 2 2 e 2 t + C

221 .

2 1 + x 2 ln | 1 + 1 + x | + C 2 1 + x 2 ln | 1 + 1 + x | + C

223 .

ln | sin x 1 sin x | + C ln | sin x 1 sin x | + C

225 .

3 4 3 4

227 .

x ln ( 1 + e x ) + C x ln ( 1 + e x ) + C

229 .

6 x 1 / 6 3 x 1 / 3 + 2 x 6 ln ( 1 + x 1 / 6 ) + C 6 x 1 / 6 3 x 1 / 3 + 2 x 6 ln ( 1 + x 1 / 6 ) + C

231 .

4 3 π arctanh [ 1 3 ] = 1 3 π ln 4 4 3 π arctanh [ 1 3 ] = 1 3 π ln 4

233 .

x = ln | t 3 | + ln | t 4 | + ln 2 x = ln | t 3 | + ln | t 4 | + ln 2

235 .

x = ln | t 1 | 2 arctan ( 2 t ) 1 2 ln ( t 2 + 1 2 ) + 2 arctan ( 2 2 ) + 1 2 ln 4.5 x = ln | t 1 | 2 arctan ( 2 t ) 1 2 ln ( t 2 + 1 2 ) + 2 arctan ( 2 2 ) + 1 2 ln 4.5

237 .

2 5 π ln 28 13 2 5 π ln 28 13

239 .

arctan [ −1 + 2 x 3 ] 3 + 1 3 ln | 1 + x | 1 6 ln | 1 x + x 2 | + C arctan [ −1 + 2 x 3 ] 3 + 1 3 ln | 1 + x | 1 6 ln | 1 x + x 2 | + C

241 .

2.0 in.2

243 .

3 ( −8 + x ) 1 / 3 3 ( −8 + x ) 1 / 3
−2 3 arctan [ −1 + ( −8 + x ) 1 / 3 3 ] −2 3 arctan [ −1 + ( −8 + x ) 1 / 3 3 ]
−2 ln [ 2 + ( −8 + x ) 1 / 3 ] −2 ln [ 2 + ( −8 + x ) 1 / 3 ]
+ ln [ 4 2 ( −8 + x ) 1 / 3 + ( −8 + x ) 2 / 3 ] + C + ln [ 4 2 ( −8 + x ) 1 / 3 + ( −8 + x ) 2 / 3 ] + C

Section 3.5 Exercises

245 .

1 2 ln | x 2 + 2 x + 2 | + 2 arctan ( x + 1 ) + C 1 2 ln | x 2 + 2 x + 2 | + 2 arctan ( x + 1 ) + C

247 .

cosh −1 ( x + 3 3 ) + C cosh −1 ( x + 3 3 ) + C

249 .

2 x 2 1 ln 2 + C 2 x 2 1 ln 2 + C

251 .

arcsin ( y 2 ) + C arcsin ( y 2 ) + C

253 .

1 2 csc ( 2 w ) + C 1 2 csc ( 2 w ) + C

255 .

9 6 2 9 6 2

257 .

2 π 2 2 π 2

259 .

1 12 tan 4 ( 3 x ) 1 6 tan 2 ( 3 x ) + 1 3 ln | sec ( 3 x ) | + C 1 12 tan 4 ( 3 x ) 1 6 tan 2 ( 3 x ) + 1 3 ln | sec ( 3 x ) | + C

261 .

2 cot ( w 2 ) 2 csc ( w 2 ) + w + C 2 cot ( w 2 ) 2 csc ( w 2 ) + w + C

263 .

1 5 ln | 2 ( 5 + 4 sin t 3 cos t ) 4 cos t + 3 sin t | 1 5 ln | 2 ( 5 + 4 sin t 3 cos t ) 4 cos t + 3 sin t |

265 .

6 x 1 / 6 3 x 1 / 3 + 2 x 6 ln [ 1 + x 1 / 6 ] + C 6 x 1 / 6 3 x 1 / 3 + 2 x 6 ln [ 1 + x 1 / 6 ] + C

267 .

x 3 cos x + 3 x 2 sin x + 6 x cos x 6 sin x + C x 3 cos x + 3 x 2 sin x + 6 x cos x 6 sin x + C

269 .

1 2 ( x 2 + ln | 1 + e x 2 | ) + C 1 2 ( x 2 + ln | 1 + e x 2 | ) + C

271 .

2 arctan ( x 1 ) + C 2 arctan ( x 1 ) + C

273 .

0.5 = 1 2 0.5 = 1 2

275 .

8.0

277 .

1 3 arctan ( 1 3 ( x + 2 ) ) + C 1 3 arctan ( 1 3 ( x + 2 ) ) + C

279 .

1 3 arctan ( x + 1 3 ) + C 1 3 arctan ( x + 1 3 ) + C

281 .

ln ( e x + 4 + e 2 x ) + C ln ( e x + 4 + e 2 x ) + C

283 .

ln x 1 6 ln ( x 6 + 1 ) arctan ( x 3 ) 3 x 3 + C ln x 1 6 ln ( x 6 + 1 ) arctan ( x 3 ) 3 x 3 + C

285 .

ln | x + 16 + x 2 | + C ln | x + 16 + x 2 | + C

287 .

1 4 cot ( 2 x ) + C 1 4 cot ( 2 x ) + C

289 .

1 2 arctan 10 1 2 arctan 10

291 .

1276.14

293 .

7.21

295 .

5 2 + ln | 2 + 2 2 1 + 5 | 5 2 + ln | 2 + 2 2 1 + 5 |

297 .

1 3 arctan ( 3 ) 0.416 1 3 arctan ( 3 ) 0.416

Section 3.6 Exercises

299 .

0.696

301 .

9.298

303 .

0.5000

305 .

T 4 = 18.75 T 4 = 18.75

307 .

0.500

309 .

1.2819

311 .

0.6577

313 .

0.0213

315 .

1.5629

317 .

1.9133

319 .

T(4) = 0.1088 T(4) = 0.1088

321 .

1.0

323 .

Approximate error is 0.000325.

325 .

1 7938 1 7938

327 .

81 25 , 000 81 25 , 000

329 .

475

331 .

174

333 .

0.1544

335 .

6.2807

337 .

4.606

339 .

3.41 ft

341 .

T16=100.125;T16=100.125; absolute error = 0.125

343 .

about 89,250 m2

345 .

parabola

Section 3.7 Exercises

347 .

divergent

349 .

π 2 π 2

351 .

2 e 2 e

353 .

Converges

355 .

Converges to 1/2.

357 .

−4

359 .

π π

361 .

diverges

363 .

diverges

365 .

1.5

367 .

diverges

369 .

diverges

371 .

diverges

373 .

Both integrals diverge.

375 .

diverges

377 .

diverges

379 .

π π

381 .

0.0

383 .

0.0

385 .

6.0

387 .

π 2 π 2

389 .

8 ln ( 16 ) 4 8 ln ( 16 ) 4

391 .

1.047 1.047

393 .

−1 + 2 3 −1 + 2 3

395 .

7.0

397 .

5 π 2 5 π 2

399 .

3 π 3 π

401 .

1 s , s > 0 1 s , s > 0

403 .

s s 2 + 4 , s > 0 s s 2 + 4 , s > 0

405 .

Answers will vary.

407 .

0.8775

Review Exercises

409 .

False

411 .

False

413 .

x 2 + 16 16 x + C x 2 + 16 16 x + C

415 .

1 10 ( 4 ln ( 2 x ) + 5 ln ( x + 1 ) 9 ln ( x + 3 ) ) + C 1 10 ( 4 ln ( 2 x ) + 5 ln ( x + 1 ) 9 ln ( x + 3 ) ) + C

417 .

4 sin 2 ( x ) sin ( x ) x 2 + C 4 sin 2 ( x ) sin ( x ) x 2 + C

419 .

1 15 ( x 2 + 2 ) 3 / 2 ( 3 x 2 4 ) + C 1 15 ( x 2 + 2 ) 3 / 2 ( 3 x 2 4 ) + C

421 .

1 16 ln ( x 2 + 2 x + 2 x 2 2 x + 2 ) 1 8 tan −1 ( 1 x ) + 1 8 tan −1 ( x + 1 ) + C 1 16 ln ( x 2 + 2 x + 2 x 2 2 x + 2 ) 1 8 tan −1 ( 1 x ) + 1 8 tan −1 ( x + 1 ) + C

423 .

M 4 = 3.312 , T 4 = 3.354 , S 4 = 3.326 M 4 = 3.312 , T 4 = 3.354 , S 4 = 3.326

425 .

M 4 = −0.982 , T 4 = −0.917 , S 4 = −0.952 M 4 = −0.982 , T 4 = −0.917 , S 4 = −0.952

427 .

approximately 0.2194

431 .

Answers may vary. Ex: 9.4059.405 km

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