Chapter 8: Integration Techniques, L'Hopital's Rule,...

Chapter 8: Integration Techniques, L'Hopital's Rule, and Improper Integrals

1. Find the indefinite integral.

( )34 – 13x dx∫ A) (x – 13)3 + C D) 4(x – 13)4 + C B) (x – 13)2 + C E) 3(x – 13)3 + C C) (x – 13)4 + C


( )62

– 9dv

v∫

A)

( )62–

5 – 9C

v+

D)

( )72

7 – 9C

v+

B)

( )72–

5 – 9C

v+

E)

( )61

3 – 9C

v+

C)

( )52–

5 – 9C

v+


( )37 +

– 2s d

s

∫ s

A)

( )

2

37–

2 2 – 2s C

s+

D)

( )

2

47+

2 4 – 2s C

s+

B)

( )

2

27–

2 2 – 2s C

s+

E)

( )

2

37+

2 3 – 2s C

s+

C)

( )

2

47–

2 2 – 2s C

s+

Copyright © Houghton Mifflin Company. All rights reserved. 229


4. Find the indefinite integral. 2

– 13p dpp∫

A) 2

169 ln – 13 + 132pp p C+ +

D) 2

169 ln – 13 + 122pp p C+ +

B) 169 ln – 13 + 12p p +C E) 2

ln – 13 + 122pp p C+ +

C) 2

ln – 13 + 132pp p C+ +


12 + 6y dy

y∫

A) 72 ln + 6 + 12y y +C D) –72 ln + 6 + 12y y +C B) 2–72 ln + 6 + 12y y +C E) 272 ln + 6 + 12y y +C C) –72 ln + 6 – 12y y +C


( )2

3

cot 7 xdx

x∫

A) ( )21 ln sin 714

x C+ D) ( )1 ln sin 7

14x C− +

B) ( )21 ln sin 77

x C− + E) ( )21 ln sin 7

14x C− +

C) ( )21 ln sin 77

x C+

7. Find the definite integral.

12

20

925x dx

x +∫

A) 117 B) 63 C) 8 D) 72 E) 7

230 Copyright © Houghton Mifflin Company. All rights reserved.


8. Find the definite integral. 3

32

0

xx e dx−∫

A) –271 13

e − D) –271 1

3e − +

B) –273 1 e − E) –27–3 1 e − C) –271 e −


23 3xx e dx∫ A) ( )23 21 3 1

9xe x − +C

D) ( )23 21 3 19

xe x C− − +

B) ( )23 21 318

xe x − +C E) ( )23 21 3

18xe x C− − +

C) ( )23 21 3 118

xe x − +C


23xx dxe∫

A) ( )23 2 2 xx x e−− + +C D) ( )2–3 2 2 xx x x e− C+ + + B) ( )2 2 2 xx x e−− + + +C E) ( )2–3 2 2 xx x e− C+ + + C) ( )2 2 2 xx x e−+ + +C


5 lnx x dx∫ A) ( )

66ln 1

25x x C − +

D) ( )6

6ln 136x x C − +

B) ( )5

5ln 136x x C − +

E) ( )6

5ln 136x x C − +

C) ( )4

4ln 136x x C − +



12. Find the indefinite integral. ( )2

2

6 ln xdx

x∫

A) ( )( )26 ln 2ln 2x xC

x

− ++

D) ( )( )26 2 ln 2xC

x

+− +

B) ( )( )26 ln 2ln 2x xC

x

+ +− +

E) ( )( )26 2 ln 2xC

x

++

C) ( )( )26 ln 2 ln 2x x C− + + +

y


( )ln + 8y y d∫ A)

( )2 2 – 64 – 16ln + 8

2 4y yy C

− +

y D)

( )2 2 – 64 – 16ln + 8

2 2y yy C

− +

y

B) ( )

2 2 – 64 – 16ln + 82 4

y yy C

+ +

y E)

( )2 2 – 64 – 8ln + 8

2 4y yy C

+ +

y

C) ( )

2 2 + 64 + 8ln + 82 4

y yy C

− +

y


3

ln7s dss∫

A) ( )2

1 2ln 114

s Cs

− + + D) ( )2

1 2ln 128

s Cs

− − +

B) ( )2

1 2 ln 1)14

s Cs

+ + E) ( )2

1 2ln 128

s Cs

− + +

C) ( )2

1 2ln 128

s Cs

− +


4 – 5p p d∫ p A) ( ) ( )3 24 – 5 6 – 5

60p p

C+ D) ( ) ( )3 24 + 5 6 – 5

60p p

C+

B) ( ) ( )3 24 – 5 6 + 1060

p pC+

E) ( ) ( )3 24 – 5 6 – 1060

p pC+

C) ( ) ( )3 24 – 5 6 + 560

p pC+




1 7n dnn+∫

A) ( )7 2 7 149

n nC

− ++

D) ( )2 7 2 7 121

n nC

− ++

B) ( )2 7 2 7 1147

n nC

− ++

E) None of the above

C) ( )2 2 7 7 1147n n

C− +

+


–3 sin 7we w∫ dw A) –3–3sin 7 + 7cos 7

58ww w e C +

D) –33sin 7 + 7cos 758

ww w e C +

B) –3–7sin 7 – 3cos 758

ww w e C +

E) –37sin 7 – 3cos 7

58ww w e C +

C) –3–3sin 7 – 7cos 758

ww w e C +


–3 cos9ue u∫ du A) –3cos9 + 3sin 9

–30uu u e C +

D) –3cos9 – 3sin 9–30

uu u e C +

B) –3– cos9 – 3sin 9–30

uu u e C +


C) –3– cos9 + 3sin 9–30

uu u e C +


/8

0

cos8x x dxπ

∫

A) 1

32− B) 1

4− C) 1

32 D) 1

2 E) 1

2−



20. Find the definite integral. 2

3

1

lnx x dx∫

A) 128ln 2 – 1516

D) 64ln 2 + 15

16

B) 64ln 2 – 1516


C) 64ln 2 – 1516

−


4cos sinx x dx∫ A) 4cos

4x C+

D) 5sin5x C− +

B) 5cos5x C− +

E) 4sin4x C+

C) 5sin5x C+


4sin cosx x dx∫ A) 5cos

5x C+

D) 5cos5x C− +

B) 5cos4x C− +

E) 5sin5x C− +

C) 5sin5x C+


3 4sin 6 cos 6x x dx∫ A) ( )2 51 7 5cos 6 cos 6

180x x C− − +

D) ( )2 51 7 5cos 6 cos 6210

x x C− +

B) ( )2 51 7 5cos 6 cos 6210

x x C− − + E) ( )2 51 7 5cos 6 cos 6

6x x C− +

C) ( )2 51 7 5cos 6 cos 66

x x C− − +



24. Find the indefinite integral. 3 2cos 2 sin 2x x dx∫

A) ( )2 4sin 2 5sin 2 3sin 210x x x C− +

D) ( )2 4sin 2 4 5sin 2 3sin 210x x x C− + +

B) ( )2 4sin 2 4 5sin 2 3sin 230x x x C− + +

E) ( )2 4sin 2 4 5sin 2 3sin 230x x x C− + +

C) ( )2 4sin 2 5sin 2 3sin 230x x x C− +


3cos 6x dx∫ A) ( )2sin 6 3 sin 6

18

x xC

−+

D) ( )4sin 6 3 sin 6

18

x xC

−+

B) ( )2sin 6 3 sin 6

3

x xC

++

E) ( )2sin 6 3 sin 6

18

x xC

++

C) ( )4sin 6 3 sin 6

18

x xC

++


4cos 5x dx∫ A) 60 8sin(5 ) sin(20 )

160x x x C+ +

+ D) 60 8sin(10 ) sin(20 )

160x x x C+ +

+

B) 20 4sin(10 ) sin(15 )160

x x x C+ ++

E) 20 8sin(10 ) sin(20 )160

x x x C+ ++

C) 60 8sin(10 ) sin(15 )160

x x x C+ ++



27. Find the indefinite integral. 3sin

7x dx∫

A) 27 3 cos cos7 7

3

x x

C

− − +

D) 27 1 cos cos7 7

3

x x

C

+ +

B) 27 2 cos cos7 7

3

x x

C

− − +

E) 27 2 cos cos7 7

3

x x

C

− +

C) 27 3 cos cos7 7

3

x x

C

− +


2sin 2x dx∫ A) 2 sin 2 cos 2

4x x x C−

+ D) 22 sin 2 cos 2

4x x x C−

+

B) 2 sin 2 cos 24

x x x C++

E) 2 sin 2 cos 22

x x x C++

C) 2 sin 2 cos 22

x x x C−+


3sin 2 cos 2 dθ θ θ∫ A) ( ) ( )3/ 2 7 / 22 2cos(2 ) cos(2 )

3 3Cθ θ− +

B) ( ) ( )3/ 2 7 / 21 1cos(2 ) cos(2 )3 7

Cθ θ− +

C) ( ) ( )3/ 2 7 / 21 1cos(2 ) cos(2 )3 7

Cθ θ− + +

D) ( ) ( )3/ 2 7 / 21 1cos(2 ) cos(2 )6 14

Cθ θ− + +

E) None of the above.



30. Find the indefinite integral. 3cos 3

sin 3dθ θ

θ∫

A) ( )2sin 3 5 sin 3

5C

θ θ−+

D) ( )22 sin 3 3 sin 3

5C

θ θ−+

B) ( )22 sin 3 5 sin 3

5C

θ θ++

E) ( )22 sin 3 5 sin 3

15C

θ θ−+

C) ( )2sin 3 3 sin 3

5C

θ θ++


3tan 9x dx∫ A) 21 1ln cos9 tan 9

9 18x x C− +

D) 21 1ln cos9 tan 9 sec99 18

x x x C+ +

B) 21 1ln cos9 tan 99 18

x x C+ + E) 21 1tan 9 ln sin 9

18 9x x C+ +

C) 21 1tan 9 ln cos918 9

x x C− +


5tan5x dx

∫

A) 2 25tan 1 tan 5ln cos5 4 5 5x x C ⋅ + − +

x

B) 2 25tan 1 tan 5ln cos5 4 5 5x x C ⋅ − − +

x

C) 2 25 1tan tan 1 5ln cos2 5 2 5 5

x x x C ⋅ − −

+

D) 2 25 5tan 1 tan 5ln cos2 5 4 5 5

x x C ⋅ − + +

x

E) 2 25tan 1 tan 5ln sin5 4 5 5x x C ⋅ + + +

x



33. Find the indefinite integral. 4sec 2x dx∫

A) ( )21 tan(2 ) 3 tan (2 )6

x x C− + D) ( )21 tan(2 ) 3 tan (2 )

6x x C+ +

B) ( )21 tan(2 ) 3 tan (2 )6

x x C− + + E) ( )1 tan(2 ) 3 tan(2 )

6x x C− +

C) ( )1 tan(2 ) 3 tan(2 )6

x x C+ +


6sec 2x dx∫ A) 3 515 tan(2 ) 10 tan(2 ) 3 tan(2 )

15x x x C+ +

+

B) 3 515 tan(2 ) 10 tan(2 ) 3 tan(2 )30

x x x C+ ++

C) 3 515 tan(2 ) 10 tan(2 ) 3 tan(2 )30

x x x C− −+

D) 3 515 tan(2 ) 10 tan(2 ) 3 tan(2 )15

x x x C− −+


35. Find the indefinite integral. 3 2tan sec

7 7x x dxπ π

∫

A) 57 tan6 7

x Cππ

+ D) 47 tan

6 7x Cπ

π+

B) 47 tan4 7

x Cππ

− + E) 47 tan

6 7x Cπ

π− +

C) 47 tan4 7

x Cππ

+


5sec 7 tan 7x x dx∫ A) 71 sec 7

35x C+

D) 51 sec 735

x C+

B) 51 sec 75

x C+ E) 61 sec 7

5x C+

C) 51 tan 7 sec 735

x x C+



37. Find the indefinite integral by making the substitution 6sinx θ= .

( )3 2236

x dxx−

∫

A) 2

136

Cx

+−

D)

2

136

Cx

− +−

B) 2

32 36

Cx

− +−


C) 2

32 36

Cx

+−


2 2

149

dxx x−∫

A) 24949

x Cx−

+ D) 2

2

4949

x Cx−

− +

B) 24949

x Cx−

− + E)

( )3 22

1

49 49C

x x+

−

C)

( )3 22

1

49 49C

x x− +

−


264 x dx

x−

∫

A) 22 8 6464 8ln xx C

x+ −

− + + D) 2

2 8 6464 8ln xx Cx

− −− − +

B) ( )

23 22 8 6464 8ln xx Cx

+ −− − +

E) ( )

23 22 8 6464 8ln xx Cx

− −− + +

C) 22 8 6464 8ln xx C

x+ −

− − +




2

281x dxx−∫

A) 2181 81sin

2 9 2x x x C− − + +

D) 2181 81cos

2 9 2x x x C− − + +

B) 2181 81sin

2 9 2x x x C− − − +

E) 2181 81cos

2 9 2x x x C− − − +

C) 2181 81cos

2 9 2x x x C− − − −

+

41. Find the indefinite integral by making the substitution 9secx θ= .

2

181dx

x −∫

A) 2ln 81x x C− + + D) 2ln 81x x C− +

B) 2ln 81x C− + E) None of the above

C) 2 2ln 81x x C+ + +


2 25x dxx−

∫

A) 2 1 525 5secx Cx

− − − +

D) 2 1 525 5secx C

x− − + +

B) 2 125 5sec5xx C− − + +

E) 2 125 25sec5xx C− − − +

C) 2 125 5sec5xx C− − − +




3

2 81x dxx −

∫

A) ( )2 281 162

2

x xC

− ++

D) ( )2 281 162

3

x xC

− ++

B) ( )2 281 81

3

x xC

− ++

E) ( )2 81 813

x xC

− ++

C) ( )2 81 1623

x xC

− ++

44. Find the indefinite integral by making the substitution 2 tanx θ= .

24x x dx+∫ A) ( )3 22 4

2x

C+

+ D) ( )3 222 4

3x

C+

− +

B) ( )3 222 363

xC

++

E) ( )3 22 43

xC

++

C) ( )3 22 43

xC

+− +

45. Find the indefinite integral by making the substitution 6 tanx θ= .

3

236x dxx+∫

A) ( )2 236 72

3

x xC

+ −+

D) ( )2 236 36

3

x xC

+ −+

B) ( )2 236 72

3

x xC

+ −+

E) ( )2 22 36 36

3

x xC

+ −+

C) ( )2 22 36 72

3

x xC

+ −+



46. Find the indefinite integral. 29 25x dx−∫

A) 219 5 9 25cos

10 3 2x x x C− − − +

+

D) 219 5 2 9 25sin

10 3 3x x x C− − + +

B) 219 5 9 25sin

10 3 3x x x C− − − +

E) 219 5 3 9 25cos

10 3 2x x x C− − − − +

C) 219 5 9 25sin

10 3 2x x x C− − + +


216 81x x dx−∫ A) ( )2 281 16 16 81

243

x xC

+ −+

D) ( )3216 81

243

xC

−− +

B) ( )3216 81

243

xC

−+


C) ( )5 2216 81243x

C−

− +


2

125 36

dxx x+∫

A) 21 36 25 55

x Cx+ −

+ D) 2

2

1 36 25 5ln10

x Cx+ −

+

B) 21 36 25 5ln5

x Cx+ +

+ E) 21 36 25 5ln

5x Cx+ −

+

C) 21 36 25 55

x Cx+ +

+




( )3 22

1

8dx

x +∫

A) 28 8x Cx

++

D) 2

28 8x Cx

++

B) 28 8x x C+ + E) 2

28 8x Cx

− ++

C) 28 8x Cx

− ++


6 2

25

25x dxx−

∫

A) ( ) 11ln 11 6 ln 56

+ − + D) ( ) 11ln 11 6

6+ −

B) ( ) 11ln 11 6 ln 56

+ − − E) ( ) 1ln 11 6 ln 5

6 11+ + −

C) ( ) 1ln 11 6 ln 56 11

+ − −

51. Write the form of the partial fraction decomposition for the following rational

expression:

2

–5 + 5x x

A) 2 +

5A Bx x

D)

2 + 5Ax B Cx x+

+

B) + 5

A Bx x+

E) 2 +

5Ax B Cx x+

C) 2 + 5A Bx x

+



52. Write the form of the partial fraction decomposition for the following rational expression:

( )

2

36 7

+ 7xx

+

A)

( ) ( )

2 3

2 3 + 7 + 7 + 7Ax B Cx D Ex Fx x x+ +

+ ++

D)

( ) ( )

2

2 37 6 + 7 + 7 + 7

x Ax x x

+ +

B)

( )

2

3 + 7Ax Cx

+ E)

( ) ( )2 3 + 7 + 7 + 7A B Cx x x

+ +

C)

( )3 + 7 + 7A Bx x

+

53. Write the form of the partial fraction decomposition for the following rational

expression:

( )22

6 7

8

x

x x

−

+

A) 8 8

A B Cx x x+ +

+ −

B)

( ) ( )2 28 88 8A B C D Ex x xx x+ + + +

+ −+ −

C)

( )22 28 8

A Bx C Dx Ex x x

+ ++ +

+ +

D)

( )22 28 8

A B Cx x x+ +

+ +

E)

( )22 8

A Bx Cx x

++

+

54.

Use partial fractions to find the integral 2

13 10913 30x dx

x x−

− +∫ .

A) 3ln 10 10ln 3x x C− + − + B) 3 30ln 10 10 30ln 3x x x x+ − + + − C+ C) 3ln 10 10ln 3x x C− − − + D) 3 30ln 10 10 30ln 3x x x x+ − − + − C+ E) 3 30ln 10 10ln 3x x x+ − + − C+



55. Use partial fractions to find the integral

3 2

2

3 – 25 + 19 + 739 14

x x x dxx x− +∫ .

A) 23 2 2ln 7 7 ln 22x x x x+ + − + − +C

D) 23 2 2ln 7 7 ln 22x x x x C+ − − + − +

B) 3 2 2ln 7 7 ln 2x x x+ + − − − +C E) 23 2 2ln 7 7 ln 22x x x x C+ − − − − +

C) 23 2 2ln 7 7 ln 22x x x x+ + − − − +C

56.


2

2 19 36( 6)

x x dxx x+ ++∫ .

A) 1ln ln 66

x x Cx

− + − ++

D) 1ln ln 6

6x x C

x+ + − +

+

B) 1ln ln 66

x x Cx

− + + ++

E) 1ln

6x C

x− +

+

C) 1ln ln 66

x x Cx

+ + + ++

57.


3

16 9 72981

x x dxx x+ ++∫ .

A) ( )279ln ln 812

x x C+ + +

B) 9745ln x Cx

− +

C) 9 ln 7 ln 9 9ln 9x x x+ + + − +C D) ( ) ( )29 ln 7 9 ln 81x x x+ + + +C E) ( )279ln ln 81 arctan

2 9xx x C + + +

+



58. Use partial fractions to find the integral

( )2

22

7 5 175

25

x x dxx

+ +

+∫ .

A) ( )22

1 5arctan ln 252( 25) 50 5 2

x x x Cx

+ + + + +

B) 7 55 5

Cx x

− − +− +

C) 2

5 7 arctan2( 25) 5 5

x Cx

− + + +

D) 2

17 arctan10( 25) 50 5

x x Cx

+ + +

E) ( ) ( )( )22 27 ln 25 5ln 25x x C+ + + +

59. Use integration tables to find ( )2arctan + 64x x dx∫ .

A) ( ) ( )( )2 22 2+ 64 1arctan + 64 ln + 64 12 4

x x x C− + +

B) ( ) ( )( )2 22 2– 64 1arctan – 64 ln – 64 12 4

x x x C− + +

C) ( ) ( )21arctan ln2 4x x x C− +

D) ( ) ( )2arctan lnx x x− +C E) ( ) ( )

2 22 2arctan + 64 ln + 64

2 4x xx x C− +

60.

Use integration tables to find 2

18 5 10

dxx x+ +∫ .

A) 1 16 5 295ln295 16 5 295

x Cx+ −

++ +

B) 2 16 5arctan295 295

u C+ +

C) 21 10 16ln 10 5 8 arctan

16 295 295x 5x x C

+ + + − +

D) 2 16 5arctan295 295

x C+ +

E) 1 16 5 295ln295 16 5 295

u Cu+ −

++ +



61. Use integration tables to find

( )2

3

61+ sin

t dtt∫ .

A) ( ) ( )3 32 tan + 2sect t +C D) ( ) ( )3 32 tan 2sect t C± + B) ( ) ( )3 32 tan – 2sect t +C E) ( ) ( )2 22 tan – 2sect t +C

C

C) ( ) ( )3 36 tan + 6sect t +

62.

Use integration tables to find ( )

81+ sec 8

x

xdx∫ .

A) ( ) ( ) ( )( )1 8 cot 8 + csc 8

ln 8x x x C+ +

D)( ) ( ) ( )( )1 cot – csc

ln 8u u u C+ +

B) ( ) ( ) ( )( )1 8 cot 8 – csc 8

ln 8x x x C+ +

E) ( ) ( ) ( )( )1 8 cot 8 csc 8

ln 8x x x C+ ± +

C) ( ) ( )8 cot 8 – csc 8x x x C+ +

63.

Use integration tables to find 2

cos( )8sin ( ) 3sin( ) 5

x dxx x+ +∫ .

A) 1 16sin( ) 3 151ln151 16sin( ) 3 151

x Cx+ −

++ +

B) 2 16cos( ) 3arctan151 151

x C+ +

C) 21 6 16sin( ) 3ln 5 3sin( ) 8sin ( ) arctan

16 151 151xx x C

+ + + −

+

D) 2 16sin( ) 3arctan151 151

x C+ +

E) 1 16cos( ) 3 151ln151 16cos( ) 3 151

x Cx+ −

++ +



64. Use integration tables to find

2 2

19 – 4

dxx x∫ .

A) 29 – 49x Cx

+ D) 29 4

4x Cx+

+

B) 29 44x Cx±

± + E) 29 – 4

4x Cx

+

C) 29 44x Cx−

− +

65. Use integration tables to find ( ) ( )2 210 9 10 9 + 36x x d+ + x∫ .

A) ( ) ( )( ) ( ) ( ) ( )2 2 21 10 9 2 10 9 +36 10 9 +36 1296 ln 10 9 10 9 +36

80x x x x x + + + − + + +

C+

B) ( ) ( )( ) ( ) ( ) ( )2 2 21 10 9 2 10 9 36 10 9 36 1296 ln 10 9 10 9 36

80x x x x x + + ± + ± − + + + ±

C+

C) ( ) ( )( ) ( ) ( ) ( )2 2 21 10 9 2 10 9 +36 10 9 +36 1296 ln 10 9 10 9 +36

8x x x x x + + + − + + +

C+

D) ( ) ( )( ) ( ) ( ) ( )2 2 21 10 9 2 10 9 –36 10 9 –36 1296 ln 10 9 10 9 –36

80x x x x x + + + − + + +

C+

E) ( ) ( )( ) ( ) ( ) ( )2 2 21 10 9 2 10 9 –36 10 9 +36 1296 ln 10 9 10 9 –36

800x x x x x + + + − + + +

C+

66.

Evaluate the limit ( )24

2 4m

16x

xx→

−−

li first by using techniques from Chapter 1 then by using

L'Hopital's Rule. A) 1–

4 B) 0 C) 1

8 D) 1

4 E) Limit does not exist.

67.

Evaluate the limit 2

9

2 26 7m9x

x xx→

− + −−

2li first by using techniques from Chapter 1 then by

using L'Hopital's Rule. A) B) 10 C) 0 D) 9 E) Limit does not exist. –10

68.


2

63 9 7lim4 9x

x xx→∞

− +−

first by using techniques from Chapter 3 then by

using L'Hopital's Rule. A) 9–

4 B) 63

4 C) 7

4 D) ∞ E) Limit does not exist.



69. Evaluate the limit

2

0

64 8m7x

xx→

− −li using L'Hopital's Rule if necessary.

A) 8

7− B) 1

7− C) 1 D) 0 E) Limit does not exist.

70.


7( 1 )lim9

x

x

ex+→

− − x using L'Hopital's Rule if necessary.

A) 7

9 B) 7

54 C) 0 D) ∞ E) –∞

71.


arcsin 11m

9xli

xx→

using L'Hopital's Rule if necessary.

A) 11

9 B) 9

11 C) 1

9 D) 0 E) Limit does not exist.

72.


1 cos(9 )m14x

li xx→

− using L'Hopital's Rule if necessary.

A) 9

14 B) 81

28 C) 9

28 D) 0 E) Limit does not exist.

73.


4lim xx

xe→∞


A) 5

4 B) 15

128 C) 1 D) 0 E) Limit does not exist.

74.

Evaluate the limit

13

4

9limx

x

ex

−

→−∞ using L'Hopital's Rule if necessary.

A) 1

216 B) C) 9 3–

4 D) –∞ E) ∞

75.


3

lnlimx

xx→∞


A) 5

3 B) 3

5 C) 0 D) −∞ E) ∞

76.


100

lnlimx

xx+→


A) 3

5 B) 5

3 C) 0 D) −∞ E) ∞



77. Evaluate the limit using L'Hopital's Rule if necessary. ( )10/

0lim 2

xx

xe x

+→+

A) B) C) D) 1 E) 30e 2e + 20e 10e

78. Evaluate the limit 28

19 24lim64 8x

x xx x+→

− − − − using L'Hopital's Rule if necessary.

A) 5

8 B) 5

16 C) 5–

16 D) 5–

8 E) Limit does not exist.

79.

Evaluate the limit 8lim 7 sinx

xx→∞


A) 8

7 B) C) 0 D) 56 ∞ E) Limit does not exist.

80. Evaluate the limit ( 2

2lim 5( 2) x

xx

+) −

→− using L'Hopital's Rule if necessary.

A) B) C) D) 0 E) 1 5e 2e e

81. Determine whether the improper integral 3

3

7 dxx

∞

∫ diverges or converges. Evaluate the

integral if it converges. A) diverges B) 7

18 C) 1

2 D) 7

27 E) 7

82.

Determine whether the improper integral / 6

0

xxe d∞

−∫ x diverges or converges. Evaluate

the integral if it converges. A) diverges B) 1

6− C) e D) E) 6 1/ 6− 36

83.

Determine whether the improper integral diverges or converges.

Evaluate the integral if it converges.

7

0

sin(6 )xe x∞

−∫ dx

A) diverges B) C) 42 6

85 D) 85 E) 6



84. Determine whether the improper integral 2

11144

dxx

∞

−∞ +∫ diverges or converges.

Evaluate the integral if it converges. A) diverges B) 11

144 C) 11

12 D) 11

144π E) 11

12π

85.

Determine whether the improper integral 1

40

11

dxx−∫ diverges or converges. Evaluate


3 C) D) 4 3

4 E) 3

86.


29

1100

dxx−∫ diverges or converges.

Evaluate the integral if it converges. A) diverges D) 9arcsin

2 1π −

0

B) 9arcsin20 100π −

E) 9arcsin2 100π −

C) 9arcsin20 10π −

87.


28

7( 9)

dxx −∫ diverges or converges. Evaluate


4 C) 7 D) 7

81 E) 81

88.


6( 81)

dxx x

∞

+∫ diverges or converges.

Evaluate the integral if it converges. A) diverges B) 2

3 C) 2

27 D) 2

3π E) 2

27π



Answer Key

1. C Section: 8.1

2. C Section: 8.1

3. B Section: 8.1

4. A Section: 8.1

5. D Section: 8.1

6. E Section: 8.1

7. D Section: 8.1

8. A Section: 8.1

9. C Section: 8.2

10. E Section: 8.2

11. D Section: 8.2

12. B Section: 8.2

13. A Section: 8.2

14. E Section: 8.2

15. C Section: 8.2

16. B Section: 8.2

17. C Section: 8.2

18. D Section: 8.2

19. A Section: 8.2

20. B Section: 8.2

21. C Section: 8.3

22. D Section: 8.3



23. B Section: 8.3

24. C Section: 8.3

25. A Section: 8.3

26. D Section: 8.3

27. A Section: 8.3

28. A Section: 8.3

29. C Section: 8.3

30. E Section: 8.3

31. B Section: 8.3

32. C Section: 8.3

33. D Section: 8.3

34. B Section: 8.3

35. C Section: 8.3

36. D Section: 8.3

37. A Section: 8.4

38. B Section: 8.4

39. D Section: 8.4

40. B Section: 8.4

41. A Section: 8.4

42. C Section: 8.4

43. D Section: 8.4

44. E Section: 8.4

45. B Section: 8.4



46. C Section: 8.4

47. D Section: 8.4

48. E Section: 8.4

49. A Section: 8.4

50. B Section: 8.4

51. B Section: 8.5

52. E Section: 8.5

53. C Section: 8.5

54. A Section: 8.5

55. C Section: 8.5

56. D Section: 8.5

57. E Section: 8.5

58. C Section: 8.5

59. A Section: 8.6

60. D Section: 8.6

61. B Section: 8.6

62. B Section: 8.6

63. D Section: 8.6

64. E Section: 8.6

65. A Section: 8.6

66. D Section: 8.7

67. A Section: 8.7

68. C Section: 8.7




69. D Section: 8.7

70. D Section: 8.7

71. A Section: 8.7

72. B Section: 8.7

73. D Section: 8.7

74. E Section: 8.7

75. C Section: 8.7

76. D Section: 8.7

77. A Section: 8.7

78. C Section: 8.7

79. B Section: 8.7

80. E Section: 8.7

81. B Section: 8.8

82. D Section: 8.8

83. C Section: 8.8

84. E Section: 8.8

85. B Section: 8.8

86. D Section: 8.8

87. A Section: 8.8

88. D Section: 8.8

Chapter 8: Integration Techniques, L'Hopital's Rule,...

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