 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.

$arcsin x - 5 5 + C arcsin x - 5 5 + C$

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.

9.484

303.

0.5000

305.

$T 4 = 18.75 T 4 = 18.75$

307.

0.500

309.

1.129

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.

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