BC3
Quiz #3
Name: ____________________
No Calculator allowed.
1. Determine whether each improper integral converges (C) or diverges (D). NO work
Necessary. Most of these should know with little or no work.

C
a)


 3
1
C
c)



1
e)
 1

 3 5
0 x

1
x4
1
dx
2
dx
e3 x
b)



0
D
d)



0
C
f)
D


1
D
dx
2. Evaluate the following integral.
2


1
 2 dx 
dx
 lim 

x  1 b1 b x  1 

2
 lim  2 x  1 
b 1
b


2
 lim  2 1  2 b  1 
b 1
b

2
1
dx
x



3
dx
x2
ex
dx
e2 x  4 x  7
3.


lim 
x  


x 1

dt 

t 
Evaluate the limit: lim  1
, Since
x   4 x  3 




x

1
1


1 t dt 

x 
  lim 

4
4 x  3     x 

 
 2 4 x  3 


 lim 
x 

1
4.
x

1

1
form. So,
dt diverges, this limit is

t
4x  3 

2 x 
Determine whether each improper integral converges or diverges. Show all steps clearly,
and/or explain reasoning completely.

a.
3x 2  4 x
 5 4 3 dx
10 2 x  x  4 x
1
3x 2  4 x
Let f ( x)  3 and g ( x)  5
.
x
2x  x4  4 3 x
f ( x)
1 2 x5  x 4  4 3 x
2 x5  x 4  4 3 x
2
 lim 3 
= lim
= = L.
Then, lim
2
5
7
x  g ( x )
x  x
x 
3
3x  4 x
3x  4 x


Since 0  L   and

1
10 x3 dx converges by p-test. It follows that
by the limit comparison test.

3x 2  4 x
10 2 x5  x4  4 3 x dx converges
4.
(continued) Determine whether each improper integral converges or diverges. Show all steps
clearly, and/or explain reasoning completely.
b.

4
0
dr
1
1
. Since 0 

for all r  (0, 4] and
r r
r r
r

test (p = 1/2). It follows that
4
0
dr
converges
r r
c.


e
1
t  ln  t  
2
dt  lim
b

b
e
1
t  ln  t  
2
dt
 1 b 
 lim 

b  ln  t  e 
 1
1 
 lim 


b  ln  b  ln  e  
 0 1
Therefore this integral converges (to 1).

1
0
dr
converges by the pr