 Calculus Volume 2

# Chapter 3

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

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 .

$2 π e 2 π e$

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 .

$−12cos2x+C−12cos2x+C$ 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 .

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 .

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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