Calculus Volume 2

# Chapter 3

### Checkpoint

3.1

$∫​xe2xdx=12xe2x−14e2x+C∫​xe2xdx=12xe2x−14e2x+C$

3.2

$12x2lnx−14x2+C12x2lnx−14x2+C$

3.3

$−x2cosx+2xsinx+2cosx+C−x2cosx+2xsinx+2cosx+C$

3.4

$π2−1π2−1$

3.5

$15sin5x+C15sin5x+C$

3.6

$13sin3x−15sin5x+C13sin3x−15sin5x+C$

3.7

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

3.8

$sinx−13sin3x+Csinx−13sin3x+C$

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

$19sec9x−17sec7x+C19sec9x−17sec7x+C$

3.13

$∫sec5xdx=14sec3xtanx−34∫sec3x∫sec5xdx=14sec3xtanx−34∫sec3x$

3.14

$∫​125sin3θdθ∫​125sin3θdθ$

3.15

$∫​32tan3θsec3θdθ∫​32tan3θsec3θdθ$

3.16

$ln|x2+x2−42|+Cln|x2+x2−42|+C$

3.17

$x−5ln|x+2|+Cx−5ln|x+2|+C$

3.18

$25ln|x+3|+35ln|x−2|+C25ln|x+3|+35ln|x−2|+C$

3.19

$x+2(x+3)3(x−4)2=Ax+3+B(x+3)2+C(x+3)3+D(x−4)+E(x−4)2x+2(x+3)3(x−4)2=Ax+3+B(x+3)2+C(x+3)3+D(x−4)+E(x−4)2$

3.20

$x2+3x+1(x+2)(x−3)2(x2+4)2=Ax+2+Bx−3+C(x−3)2+Dx+Ex2+4+Fx+G(x2+4)2x2+3x+1(x+2)(x−3)2(x2+4)2=Ax+2+Bx−3+C(x−3)2+Dx+Ex2+4+Fx+G(x2+4)2$

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 $∫e+∞1xdx=+∞,∫e+∞1xdx=+∞,$ $∫e+∞lnxxdx∫e+∞lnxxdx$ diverges.

### Section 3.1 Exercises

1.

$u=x3u=x3$

3.

$u=y3u=y3$

5.

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

7.

$−x+xlnx+C−x+xlnx+C$

9.

$xtan−1x−12ln(1+x2)+Cxtan−1x−12ln(1+x2)+C$

11.

$−12xcos(2x)+14sin(2x)+C−12xcos(2x)+14sin(2x)+C$

13.

$e−x(−1−x)+Ce−x(−1−x)+C$

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.

$−e−x22+C−e−x22+C$

23.

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

25.

$2x−2xlnx+x(lnx)2+C2x−2xlnx+x(lnx)2+C$

27.

$(−x39+13x3lnx)+C(−x39+13x3lnx)+C$

29.

$−121−4x2+xcos−1(2x)+C−121−4x2+xcos−1(2x)+C$

31.

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

33.

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

35.

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

37.

$−coshx+xsinhx+C−coshx+xsinhx+C$

39.

$14−34e214−34e2$

41.

2

43.

$2π2π$

45.

$−2+π−2+π$

47.

$−sin(x)+ln[sin(x)]sinx+C−sin(x)+ln[sin(x)]sinx+C$

49.

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 $u=x2−3,u=x2−3,$ 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.

61.

$2πe2πe$

63.

2.05

65.

$12π12π$

67.

$8π28π2$

### Section 3.2 Exercises

69.

$cos2xcos2x$

71.

$1−cos(2x)21−cos(2x)2$

73.

$sin4x4+Csin4x4+C$

75.

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

77.

$sec2(x2)+Csec2(x2)+C$

79.

$–cosx+13cos2x+C–cosx+13cos2x+C$

81.

$−12cos2x+C−12cos2x+C$ or $12sin2x+C12sin2x+C$

83.

$–13cos3x+25cos5x–17cos7x+C–13cos3x+25cos5x–17cos7x+C$

85.

$23(sinx)32+C23(sinx)32+C$

87.

$secx+Csecx+C$

89.

$12secxtanx−12ln(secx+tanx)+C12secxtanx−12ln(secx+tanx)+C$

91.

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

93.

$−ln|cotx+cscx|+C−ln|cotx+cscx|+C$

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.

$3θ8−14πsin(2πθ)+132πsin(4πθ)+C=f(x)3θ8−14πsin(2πθ)+132πsin(4πθ)+C=f(x)$

119.

$ln(3)ln(3)$

121.

$∫−ππsin(2x)cos(3x)dx=0∫−ππsin(2x)cos(3x)dx=0$

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.

$4(x−12)24(x−12)2$

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.

$−1−x2x+C−1−x2x+C$

141.

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

143.

$−139−θ2(18+θ2)+C−139−θ2(18+θ2)+C$

145.

$(−1+x2)(2+3x2)x6−x815x3+C(−1+x2)(2+3x2)x6−x815x3+C$

147.

$−x9−9+x2+C−x9−9+x2+C$

149.

$12(ln|x+x2−1|+xx2−1)+C12(ln|x+x2−1|+xx2−1)+C$

151.

$−1+x2x+C−1+x2x+C$

153.

$18(x(5−2x2)1−x2+3arcsinx)+C18(x(5−2x2)1−x2+3arcsinx)+C$

155.

$lnx−ln|1+1−x2|+Clnx−ln|1+1−x2|+C$

157.

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

159.

$−1+x2x+arcsinhx+C−1+x2x+arcsinhx+C$

161.

$−11+x+C−11+x+C$

163.

$2−10+xxln|−10+x+x|(10−x)x+C2−10+xxln|−10+x+x|(10−x)x+C$

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.

$π28+π4π28+π4$

177.

$y=116ln|x+8x−8|+3y=116ln|x+8x−8|+3$

179.

24.6 m3

181.

$2π32π3$

### Section 3.4 Exercises

183.

$−2x+1+52(x+2)+12x−2x+1+52(x+2)+12x$

185.

$1x2+3x1x2+3x$

187.

$2x2+4x+8+16x−22x2+4x+8+16x−2$

189.

$−1x2−1x+1x−1−1x2−1x+1x−1$

191.

$−12(x−2)+12(x−1)−16x+16(x−3)−12(x−2)+12(x−1)−16x+16(x−3)$

193.

$1x−1+2x+1x2+x+11x−1+2x+1x2+x+1$

195.

$2x+1+xx2+4−1(x2+4)22x+1+xx2+4−1(x2+4)2$

197.

$−ln|2−x|+2ln|4+x|+C−ln|2−x|+2ln|4+x|+C$

199.

$12ln|4−x2|+C12ln|4−x2|+C$

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.

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

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.

$−3x+4ln|x+2|+x+C−3x+4ln|x+2|+x+C$

211.

$−ln|3−x|+12ln|x2+4|+C−ln|3−x|+12ln|x2+4|+C$

213.

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

215.

$−x+ln|1−ex|+C−x+ln|1−ex|+C$

217.

$15ln|cosx+3cosx−2|+C15ln|cosx+3cosx−2|+C$

219.

$12−2e2t+C12−2e2t+C$

221.

$21+x−2ln|1+1+x|+C21+x−2ln|1+1+x|+C$

223.

$ln|sinx1−sinx|+Cln|sinx1−sinx|+C$

225.

$3434$

227.

$x−ln(1+ex)+Cx−ln(1+ex)+C$

229.

$6x1/6−3x1/3+2x−6ln(1+x1/6)+C6x1/6−3x1/3+2x−6ln(1+x1/6)+C$

231.

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

233.

$x=−ln|t−3|+ln|t−4|+ln2x=−ln|t−3|+ln|t−4|+ln2$

235.

$x=ln|t−1|−2arctan(2t)−12ln(t2+12)+2arctan(22)+12ln4.5x=ln|t−1|−2arctan(2t)−12ln(t2+12)+2arctan(22)+12ln4.5$

237.

$25πln281325πln2813$

239.

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

241.

2.0 in.2

243.

$3(−8+x)1/33(−8+x)1/3$
$−23arctan[−1+(−8+x)1/33]−23arctan[−1+(−8+x)1/33]$
$−2ln[2+(−8+x)1/3]−2ln[2+(−8+x)1/3]$
$+ln[4−2(−8+x)1/3+(−8+x)2/3]+C+ln[4−2(−8+x)1/3+(−8+x)2/3]+C$

### Section 3.5 Exercises

245.

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

247.

$cosh−1(x+33)+Ccosh−1(x+33)+C$

249.

$2x2−1ln2+C2x2−1ln2+C$

251.

$arcsin(y2)+Carcsin(y2)+C$

253.

$−12csc(2w)+C−12csc(2w)+C$

255.

$9−629−62$

257.

$2−π22−π2$

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.

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

265.

$6x1/6−3x1/3+2x−6ln[1+x1/6]+C6x1/6−3x1/3+2x−6ln[1+x1/6]+C$

267.

$−x3cosx+3x2sinx+6xcosx−6sinx+C−x3cosx+3x2sinx+6xcosx−6sinx+C$

269.

$12(x2+ln|1+e−x2|)+C12(x2+ln|1+e−x2|)+C$

271.

$2arctan(x−1)+C2arctan(x−1)+C$

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.

$lnx−16ln(x6+1)−arctan(x3)3x3+Clnx−16ln(x6+1)−arctan(x3)3x3+C$

285.

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

287.

$−14cot(2x)+C−14cot(2x)+C$

289.

$12arctan1012arctan10$

291.

1276.14

293.

7.21

295.

$5−2+ln|2+221+5|5−2+ln|2+221+5|$

297.

$13arctan(3)≈0.41613arctan(3)≈0.416$

### Section 3.6 Exercises

299.

0.696

301.

9.298

303.

0.5000

305.

$T4=18.75T4=18.75$

307.

0.500

309.

1.2819

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.

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.

407.

0.8775

### Chapter Review Exercises

409.

False

411.

False

413.

$−x2+1616x+C−x2+1616x+C$

415.

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

417.

$−4−sin2(x)sin(x)−x2+C−4−sin2(x)sin(x)−x2+C$

419.

$115(x2+2)3/2(3x2−4)+C115(x2+2)3/2(3x2−4)+C$

421.

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

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