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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. Key Terms
    10. Key Equations
    11. Key Concepts
    12. Chapter 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. Key Terms
    12. Key Equations
    13. Key Concepts
    14. Chapter 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. Key Terms
    10. Key Equations
    11. Key Concepts
    12. Chapter 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. Key Terms
    8. Key Equations
    9. Key Concepts
    10. Chapter 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. Key Terms
    9. Key Equations
    10. Key Concepts
    11. Chapter 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. Key Terms
    7. Key Equations
    8. Key Concepts
    9. Chapter 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. Key Terms
    8. Key Equations
    9. Key Concepts
    10. Chapter 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

xe2xdx=12xe2x14e2x+Cxe2xdx=12xe2x14e2x+C

3.2

12x2lnx14x2+C12x2lnx14x2+C

3.3

x2cosx+2xsinx+2cosx+Cx2cosx+2xsinx+2cosx+C

3.4

π21π21

3.5

15sin5x+C15sin5x+C

3.6

13sin3x15sin5x+C13sin3x15sin5x+C

3.7

12x+14sin(2x)+C12x+14sin(2x)+C

3.8

sinx13sin3x+Csinx13sin3x+C

3.9

12x+112sin(6x)+C12x+112sin(6x)+C

3.10

12sinx+122sin(11x)+C12sinx+122sin(11x)+C

3.11

16tan6x+C16tan6x+C

3.12

19sec9x17sec7x+C19sec9x17sec7x+C

3.13

sec5xdx=14sec3xtanx34sec3xsec5xdx=14sec3xtanx34sec3x

3.14

125sin3θdθ125sin3θdθ

3.15

32tan3θsec3θdθ32tan3θsec3θdθ

3.16

ln|x2+x242|+Cln|x2+x242|+C

3.17

x5ln|x+2|+Cx5ln|x+2|+C

3.18

25ln|x+3|+35ln|x2|+C25ln|x+3|+35ln|x2|+C

3.19

x+2(x+3)3(x4)2=Ax+3+B(x+3)2+C(x+3)3+D(x4)+E(x4)2x+2(x+3)3(x4)2=Ax+3+B(x+3)2+C(x+3)3+D(x4)+E(x4)2

3.20

x2+3x+1(x+2)(x3)2(x2+4)2=Ax+2+Bx3+C(x3)2+Dx+Ex2+4+Fx+G(x2+4)2x2+3x+1(x+2)(x3)2(x2+4)2=Ax+2+Bx3+C(x3)2+Dx+Ex2+4+Fx+G(x2+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

24352435

3.23

17241724

3.24

0.0074, 1.1%

3.25

11921192

3.26

25362536

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=x3u=x3

3.

u=y3u=y3

5.

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

7.

x+xlnx+Cx+xlnx+C

9.

xtan−1x12ln(1+x2)+Cxtan−1x12ln(1+x2)+C

11.

12xcos(2x)+14sin(2x)+C12xcos(2x)+14sin(2x)+C

13.

ex(−1x)+Cex(−1x)+C

15.

2xcosx+(−2+x2)sinx+C2xcosx+(−2+x2)sinx+C

17.

12(1+2x)(−1+ln(1+2x))+C12(1+2x)(−1+ln(1+2x))+C

19.

12ex(cosx+sinx)+C12ex(cosx+sinx)+C

21.

ex22+Cex22+C

23.

12xcos[ln(2x)]+12xsin[ln(2x)]+C12xcos[ln(2x)]+12xsin[ln(2x)]+C

25.

2x2xlnx+x(lnx)2+C2x2xlnx+x(lnx)2+C

27.

(x39+13x3lnx)+C(x39+13x3lnx)+C

29.

1214x2+xcos−1(2x)+C1214x2+xcos−1(2x)+C

31.

(−2+x2)cosx+2xsinx+C(−2+x2)cosx+2xsinx+C

33.

x(−6+x2)cosx+3(−2+x2)sinx+Cx(−6+x2)cosx+3(−2+x2)sinx+C

35.

12x(11x2+x·sec−1x)+C12x(11x2+x·sec−1x)+C

37.

coshx+xsinhx+Ccoshx+xsinhx+C

39.

1434e21434e2

41.

2

43.

2π2π

45.

−2+π−2+π

47.

sin(x)+ln[sin(x)]sinx+Csin(x)+ln[sin(x)]sinx+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.

2πe2πe

63.

2.05

65.

12π12π

67.

8π28π2

Section 3.2 Exercises

69.

cos2xcos2x

71.

1cos(2x)21cos(2x)2

73.

sin4x4+Csin4x4+C

75.

112tan6(2x)+C112tan6(2x)+C

77.

sec2(x2)+Csec2(x2)+C

79.

3cosx4+112cos(3x)+C=cosx+cos3x3+C3cosx4+112cos(3x)+C=cosx+cos3x3+C

81.

12cos2x+C12cos2x+C

83.

5cosx641192cos(3x)+5cosx641192cos(3x)+ 3320cos(5x)1448cos(7x)+C3320cos(5x)1448cos(7x)+C

85.

23(sinx)2/3+C23(sinx)2/3+C

87.

secx+Csecx+C

89.

12secxtanx12ln(secx+tanx)+C12secxtanx12ln(secx+tanx)+C

91.

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

93.

ln|cotx+cscx|+Cln|cotx+cscx|+C

95.

sin3(ax)3a+Csin3(ax)3a+C

97.

π2π2

99.

x2+112sin(6x)+Cx2+112sin(6x)+C

101.

x+Cx+C

103.

0

105.

0

107.

0

109.

Approximately 0.239

111.

22

113.

1.0

115.

0

117.

3θ814πsin(2πθ)+132πsin(4πθ)+C=f(x)3θ814πsin(2πθ)+132πsin(4πθ)+C=f(x)

119.

ln(3)ln(3)

121.

ππsin(2x)cos(3x)dx=0ππsin(2x)cos(3x)dx=0

123.

tan(x)x(8tanx21+27secx2tanx)+C=f(x)tan(x)x(8tanx21+27secx2tanx)+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.

9tan2θ9tan2θ

129.

a2cosh2θa2cosh2θ

131.

4(x12)24(x12)2

133.

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

135.

ln|x+a2+x2|+Cln|x+a2+x2|+C

137.

13ln|9x2+1+3x|+C13ln|9x2+1+3x|+C

139.

1x2x+C1x2x+C

141.

9[xx2+918+12ln|x2+93+x3|]+C9[xx2+918+12ln|x2+93+x3|]+C

143.

139θ2(18+θ2)+C139θ2(18+θ2)+C

145.

(−1+x2)(2+3x2)x6x815x3+C(−1+x2)(2+3x2)x6x815x3+C

147.

x9−9+x2+Cx9−9+x2+C

149.

12(ln|x+x21|+xx21)+C12(ln|x+x21|+xx21)+C

151.

1+x2x+C1+x2x+C

153.

18(x(52x2)1x2+3arcsinx)+C18(x(52x2)1x2+3arcsinx)+C

155.

lnxln|1+1x2|+Clnxln|1+1x2|+C

157.

−1+x2x+ln|x+−1+x2|+C−1+x2x+ln|x+−1+x2|+C

159.

1+x2x+arcsinhx+C1+x2x+arcsinhx+C

161.

11+x+C11+x+C

163.

2−10+xxln|−10+x+x|(10x)x+C2−10+xxln|−10+x+x|(10x)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.

π28+π4π28+π4

177.

y=116ln|x+8x8|+3y=116ln|x+8x8|+3

179.

24.6 m3

181.

2π32π3

Section 3.4 Exercises

183.

2x+1+52(x+2)+12x2x+1+52(x+2)+12x

185.

1x2+3x1x2+3x

187.

2x2+4x+8+16x22x2+4x+8+16x2

189.

1x21x+1x11x21x+1x1

191.

12(x2)+12(x1)16x+16(x3)12(x2)+12(x1)16x+16(x3)

193.

1x1+2x+1x2+x+11x1+2x+1x2+x+1

195.

2x+1+xx2+41(x2+4)22x+1+xx2+41(x2+4)2

197.

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

199.

12ln|4x2|+C12ln|4x2|+C

201.

2(x+13arctan(1+x3))+C2(x+13arctan(1+x3))+C

203.

2ln|x|3ln|1+x|+C2ln|x|3ln|1+x|+C

205.

116(4−2+xln|−2+x|+ln|2+x|)+C116(4−2+xln|−2+x|+ln|2+x|)+C

207.

130(−25arctan[1+x5]+2ln|−4+x|ln|6+2x+x2|)+C130(−25arctan[1+x5]+2ln|−4+x|ln|6+2x+x2|)+C

209.

3x+4ln|x+2|+x+C3x+4ln|x+2|+x+C

211.

ln|3x|+12ln|x2+4|+Cln|3x|+12ln|x2+4|+C

213.

ln|x2|12ln|x2+2x+2|+Cln|x2|12ln|x2+2x+2|+C

215.

x+ln|1ex|+Cx+ln|1ex|+C

217.

15ln|cosx+3cosx2|+C15ln|cosx+3cosx2|+C

219.

122e2t+C122e2t+C

221.

21+x2ln|1+1+x|+C21+x2ln|1+1+x|+C

223.

ln|sinx1sinx|+Cln|sinx1sinx|+C

225.

3434

227.

xln(1+ex)+Cxln(1+ex)+C

229.

6x1/63x1/3+2x6ln(1+x1/6)+C6x1/63x1/3+2x6ln(1+x1/6)+C

231.

43πarctanh[13]=13πln443πarctanh[13]=13πln4

233.

x=ln|t3|+ln|t4|+ln2x=ln|t3|+ln|t4|+ln2

235.

x=ln|t1|2arctan(2t)12ln(t2+12)+2arctan(22)+12ln4.5x=ln|t1|2arctan(2t)12ln(t2+12)+2arctan(22)+12ln4.5

237.

25πln281325πln2813

239.

arctan[−1+2x3]3+13ln|1+x|16ln|1x+x2|+Carctan[−1+2x3]3+13ln|1+x|16ln|1x+x2|+C

241.

2.0 in.2

243.

3(−8+x)1/33(−8+x)1/3
−23arctan[−1+(−8+x)1/33]−23arctan[−1+(−8+x)1/33]
−2ln[2+(−8+x)1/3]−2ln[2+(−8+x)1/3]
+ln[42(−8+x)1/3+(−8+x)2/3]+C+ln[42(−8+x)1/3+(−8+x)2/3]+C

Section 3.5 Exercises

245.

12ln|x2+2x+2|+2arctan(x+1)+C12ln|x2+2x+2|+2arctan(x+1)+C

247.

cosh−1(x+33)+Ccosh−1(x+33)+C

249.

2x21ln2+C2x21ln2+C

251.

arcsin(y2)+Carcsin(y2)+C

253.

12csc(2w)+C12csc(2w)+C

255.

962962

257.

2π22π2

259.

112tan4(3x)16tan2(3x)+13ln|sec(3x)|+C112tan4(3x)16tan2(3x)+13ln|sec(3x)|+C

261.

2cot(w2)2csc(w2)+w+C2cot(w2)2csc(w2)+w+C

263.

15ln|2(5+4sint3cost)4cost+3sint|15ln|2(5+4sint3cost)4cost+3sint|

265.

6x1/63x1/3+2x6ln[1+x1/6]+C6x1/63x1/3+2x6ln[1+x1/6]+C

267.

x3cosx+3x2sinx+6xcosx6sinx+Cx3cosx+3x2sinx+6xcosx6sinx+C

269.

12(x2+ln|1+ex2|)+C12(x2+ln|1+ex2|)+C

271.

2arctan(x1)+C2arctan(x1)+C

273.

0.5=120.5=12

275.

8.0

277.

13arctan(13(x+2))+C13arctan(13(x+2))+C

279.

13arctan(x+13)+C13arctan(x+13)+C

281.

ln(ex+4+e2x)+Cln(ex+4+e2x)+C

283.

lnx16ln(x6+1)arctan(x3)3x3+Clnx16ln(x6+1)arctan(x3)3x3+C

285.

ln|x+16+x2|+Cln|x+16+x2|+C

287.

14cot(2x)+C14cot(2x)+C

289.

12arctan1012arctan10

291.

1276.14

293.

7.21

295.

52+ln|2+221+5|52+ln|2+221+5|

297.

13arctan(3)0.41613arctan(3)0.416

Section 3.6 Exercises

299.

0.696

301.

9.279

303.

0.5000

305.

T4=18.75T4=18.75

307.

0.500

309.

1.1614

311.

0.6577

313.

0.0213

315.

1.5629

317.

1.9133

319.

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

321.

1.0

323.

Approximate error is 0.000325.

325.

1793817938

327.

8125,0008125,000

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.

π2π2

351.

2e2e

353.

Converges

355.

Converges to 1/2.

357.

−4

359.

ππ

361.

diverges

363.

diverges

365.

1.5

367.

diverges

369.

diverges

371.

diverges

373.

Both integrals diverge.

375.

diverges

377.

diverges

379.

ππ

381.

0.0

383.

0.0

385.

6.0

387.

π2π2

389.

8ln(16)48ln(16)4

391.

1.0471.047

393.

−1+23−1+23

395.

7.0

397.

5π25π2

399.

3π3π

401.

1s,s>01s,s>0

403.

ss2+4,s>0ss2+4,s>0

405.

Answers will vary.

407.

0.8775

Chapter Review Exercises

409.

False

411.

False

413.

x2+1616x+Cx2+1616x+C

415.

110(4ln(2x)+5ln(x+1)9ln(x+3))+C110(4ln(2x)+5ln(x+1)9ln(x+3))+C

417.

4sin2(x)sin(x)x2+C4sin2(x)sin(x)x2+C

419.

115(x2+2)3/2(3x24)+C115(x2+2)3/2(3x24)+C

421.

116ln(x2+2x+2x22x+2)18tan−1(1x)+18tan−1(x+1)+C116ln(x2+2x+2x22x+2)18tan−1(1x)+18tan−1(x+1)+C

423.

M4=3.312,T4=3.354,S4=3.326M4=3.312,T4=3.354,S4=3.326

425.

M4=−0.982,T4=−0.917,S4=−0.952M4=−0.982,T4=−0.917,S4=−0.952

427.

approximately 0.2194

431.

Answers may vary. Ex: 9.4059.405 km

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