Thomas Calculus 13th [Solutions]

622 Chapter 8 Techniques of Integration
48.
dx
4 ;
x 1
1
x 1
x
x x x 1 x
1
x
1 1
1 1 0
lim lim lim 1 and
1
x
dx
b
4 x b
4
lim 2 x , which diverges
dx
4 x 1
diverges by the Limit Comparison Test.
49.
dv
2 ;
v 1
1
v 1
v
v v v v
1
v
1 1
lim lim lim 1 and
1 1
1 1 0
v
dv
b
2 v b
2
lim 2 v , which diverges
dv
2 v 1
diverges by the Limit Comparison Test.
50.
d
0 1 e
;
d
0 1 e
0
1 1
1 e e
for 0 and
by the Direct Comparison Test.
b
d
b
lim lim 1 1
d
0 e b
0 b
0 e
e e converges
dx
1
dx dx
1
dx dx
x 1 x 1 x 1 x 1 x
51.
3
0 6 0 6 1 6 0 6 1
dx
0 6
x 1
converges by the Direct Comparison Test.
dx
1
b
1 1 1
1 x b 2x 1 b 2b
2 2
and lim
lim
3 2 2
52.
dx
2 2
x 1
;
1
x
2 1
x
1
1 2 1
x
x 1 1
x
2
lim lim lim 1;
x x x
2
1
b
dx lim ln b , which diverges
x
b
2
dx
2 2
x 1
diverges by the Limit Comparison Test.
x 1
53. dx;
1
2
x
x
x
2
x
lim lim lim 1 1;
x x x 1 x 1
x 1
x
2
x 1
2 dx converges by the Limit Comparison Test.
1
2
x
1
x
x
1/2
dx dx lim 2x
b
lim 2 2
1
2
1
3/2
x x b 1 b b
54.
x dx
2 4
1
x
;
x dx
2 4
x 1
x
4 4
x 1 x
1
x
x
4 1 1
1
4
4
x
x
lim lim lim 1;
x x x
diverges by the Limit Comparison Test.
x dx dx
2 x 2 x
4 2
b
lim ln x , which diverges
x
55.
2 cos x
dx;
1 2 cos x
0
for x and dx
b
lim ln x , which diverges 2 cos x dx
x
x x
x
b
x
diverges by the Direct Comparison Test.
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