Thomas Calculus 13th [Solutions]

Section 8.8 Improper Integrals 619
18.
2 2 1
2
1
c
dx dx dx lim dx
c
lim dx lim sec | x| lim sec | x|
x x 1 x x 1 x x 1 b 1 x x 1 c x x 1 b 1
b c
1 2 1 2 2 2 b 2 2 2
2
1 1 1 1
lim sec 2 sec b lim sec c sec 2 0
3 2 3 2
b 1
c
19. dv
1 1 1
0 lim ln 1 tan v
b
lim ln 1 tan b ln 1 tan 0 ln 1 2 1
1 1 tan 0
2
ln(1 0)
v v b b
ln 1
2
20.
21.
22.
23.
1
2
b
2 2
16 tan x
1 1 1
2
2
dx
0 lim 8 tan x lim 8 tan b 8 tan 0 8 2
1
2
8(0) 2
x b
0
b
0 0
0 0 b b
e d lim e e lim 0 e e be e 1 lim b 1 1 lim 1
b
b b b b e
b e
1 0 1
(l'Hopital's rule for
form)
0 2 sin lim b
e d
0 2 e sin d lim 2 e
1 1
( sin cos ) b
(Formula 107 with a 1, b 1)
b
b
0
lim 2(sin b cos b) 2(sin 0 cos0) 0 2(0 1)
0
2 2
2
1
b
b
e
e
0 x 0 x x
0
b
e dx e dx lim e lim 1 e (1 0) 1
b b b
b
24.
0
0
x x x x x
2xe dx 2xe dx 2xe dx lim e lim e
0
b
b c
2 2 2 2 2
2 2
b
c
lim 1 e lim e ( 1) ( 1 0) (0 1) 0
b
c
c
0
25.
1 2 2 1
2 2
1 1 1 ln
0 ln lim 2 ln 4 lim 2 ln1 4 2 ln 4 4
lim b
x x dx x x x b b b
0
b 0 b b 0 b 0
1
b
1 lim 1 lim b 1 0 1
4 4 4 4 4
b 0 b 0
4
b
3
2
2
b
2
26.
27.
1 1
ln
ln x dx lim x x ln x b
lim 1 1ln1 ln 1 0 lim 1 lim b
b
b b b
0 1
b 0 b 0 b 0 b b 0
1 lim b 1 0 1
b 0
2 1
b
ds 1 1
lim sin s lim sin b sin 0 0
4 s
2 2 2 2
b 2 b 2
0 2
0
1
1
b
2
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