Thomas Calculus 13th [Solutions]

Chapter 8 Practice Exercises 643
d
1
d
2
d d converges if each integral converges, but
2 ( 1) 2 ( 1) 1 ( 1) 2 ( 1)
56.
3/5 3/5 3/5 3/5
3/5
lim 1
3/5
( 1)
and d diverges d diverges
2
3/5
2
3/5
( 1)
2 du b
57. du du lim ln u 2 lim ln b 2 ln 3 2 0 ln 1 ln 3
3
2
u 2u 3 u 2 3 u
b
u 3 b
b 3 3
3v
1 1 1 4 1
b
dv dv lim ln v ln(4v
1) limb
ln 1 (ln1 1 ln 3)
1 4v v 1 v v 4v 1
b
v 1 b
4b 1 b
58.
3 2 2
ln 1 1 ln 3 1 ln 3
4 4
59.
60.
2 x 2 x x x
b
2 b b b
x e dx lim x e 2xe 2e lim b e 2be 2 e
0 b
0 b
( 2) 0 2 2
0 3x 3x 3
0
1 x 1 3b 1 3b
xe dx lim x e e lim b e e 1 0 1
b
3 9 b b
9 3 9 9 9
dx 2 dx 1 dx 1 2 1 2 1 2 1 2 2 1
2 2 lim 4 9 4 9 3 tan x b
3 0 2 lim 3 tan b
3 3
tan (0)
x x x b b
61.
2
0
2
0
2 9
4
1 2 0
2 3 2 6
62. 4 2 4
1 1 1
2
0
2 lim tan b
dx dx x
2
16 16 4 2 lim tan b
0
4 tan (0) 2 2
0
x x b b
63.
lim
2 1
1 and
6
d diverges
d diverges
6 2
1
64.
65.
u u
b
u u
b
u
I e cosu du lim e cos u e sin u du 1 lim e sin u e cos u du
0 b
0 0 b
0 0
I 1 0 I 2I 1 I 1 converges
2
2 e
2 b
2
2
ln z
e
ln z ln z (ln z) (ln z) 1 (ln b)
lim 0 lim
1
1 z dz 1 z dz e z
dz 2 2 2 2 2
diverges
1 b e
b
66. 0
t
e
t
t
t
e for t 1 and e 1
dt converges
1
t
e
t
dt converges
67.
2 dx 2 4
2
dx dx
x x
0
x x
0
x
e e e e e
converges 2dx
x
e e
x
converges
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