Chapter 6
93.
Which of the following are appropriate for integration by substitution? (Check all that
apply.)
A)
3
25x
x e dx
B)
2
44
tdt
t+
C)
3sin d
 
D)
2
65x x dx+
E)
94.
Find
4
( 3)x dx+
using integration by substitution.
A)
5
1( 3)
5xC++
B)
5
( 3)xC++
C)
5
1( 3)
3xC++
D)
5
13
5xC++
95.
Find
45
sin( )t t dt
using integration by substitution.
A)
5
1cos( )
5tC+
B)
5
1cos( )
5tC−+
C)
55
1cos( )
5t t C+
D)
55
1cos( )
5t t C−+
Chapter 6
96.
Find
28
ydy
y+
using integration by substitution.
A)
3
3
8
yC
yy
+
+
B)
( )
2
8 ln 8y y C++
C)
( )
2
1ln 8
2yC++
D)
97.
Find
2
cos (sin 6) d
 
+
using integration by substitution.
A)
3
sin (sin 6)
3C
++
B)
3
1(sin 6)
6C
++
C)
3
(sin 6) C
++
D)
3
1(sin 6)
3C
++
98.
Find
2
5x
xe dx
using integration by substitution.
A)
2
5
1
10
x
eC+
B)
2
5
1
5
x
eC+
C)
3
5
1
3
x
eC+
D)
3
5
1
15
x
eC+
Chapter 6
99.
Find
23
4z z dz+
using integration by substitution.
A)
3 3/ 2
2( 4)
3zC++
B)
3 3/ 2
2( 4)
9zC++
C)
3 3/ 2
1( 4)
6zC++
D)
2
3 3/ 2
( 4)
6
zzC++
100.
Find
23
10 ( 10)y y dy+
.
A)
24
5( 10)
2yC++
B)
24
5( 10)
4
yyC++
C)
24
5( 10)
4yC++
D)
24
5( 10)
2
yyC++
101.
Find
2
2
16
xdx
x−
.
A)
2
2 3/ 2
3
(16 )
xC
x+
−
B)
2 1/ 2
2 (16 )x x C−+
C)
2 1/ 2
3(16 )xC− − +
D)
2 1/ 2
2(16 )xC− − +
Chapter 6
102.
Which of the following is equivalent to
26
( ) ( 1)f x x C= + +
?
A)
25
12 ( 1)x x dx+
B)
25
6 ( 1)x x dx+
C)
27
12 ( 1)x x dx+
D)
27
6 ( 1)x x dx+
difficulty: medium section: 6.6
103.
Which of the following is equivalent to
7
( ) ln( 1)f x x C= + +
?
A)
6
8
7
8
xdx
xx+
B)
6
7
7
1
xdx
x+
C)
7
7
ln( 1)
1
xdx
x
+
+
D)
7
8
ln( 1)
8
xdx
xx
+
+
difficulty: hard section: 6.6
104.
22
( 1)x x dx+
=
642
6 2 2
x x x C+++
A)
True
B)
False
difficulty: easy section: 6.6
Chapter 6
105.
Find
2
11 cos( )x x dx
.
A)
2
11sin( )xC+
B)
2
11sin( )
2xC+
C)
2
11sin( )xC−+
D)
2
11sin( )
2xC−+
106.
Evaluate
2
9
xdx
x−
.
A)
2
9xC− − +
B)
2
9xC−+
C)
2
19
3xC− − +
D)
2
19
9xC−+
107.
Consider
1
1/ 2 cos(2 )x dx
. What is the definite integral obtained after making the
substitution
2wx=
?
A)
2
1
1cos
2wdw
B)
2
1cos wdw
C)
1
1/ 2
1cos
2wdw
D)
1
1/ 2 cos wdw
Chapter 6
108.
Find
( )
3
2
77y y dy+
A)
( )
4
2
77
8yC
y++
B)
( )
24
2
77
8
yyC
++
C)
( )
4
2
77
8yC++
D)
( )
4
2
77
4yC++
109.
Find
2
2
6
xdx
x
−
A)
( )
3/ 2
2
4
36
C
x
−+
−
B)
( )
3/ 2
2
4
36
C
x
+
−
C)
2
26 xC−+
D)
2
26 xC− − +
difficulty: medium section: 6.6
110.
Fuel pressure in the fuel tanks of the space shuttle is decreasing at a rate of
0.1
( ) 15 t
r t e−
=
psi per second at time t in seconds. At what rate, in psi/sec, is pressure
decreasing at 15 seconds? Round to 2 decimal places.
Ans:
3.35 psi/sec
Chapter 6
111.
Calculate
2
63z
ze dz
+
.
A)
2
63
6
z
zeC
++
B)
2
63
12
z
zeC
++
C)
2
63
1
6
z
eC
++
D)
2
63
1
12
z
eC
++
112.
Suppose
‘( ) 3x
Fx=
and
(0) 6F=
. Find
(1.5)F
to 2 decimal places.
113.
The following figure shows the graph of
()fx
. If
‘Ff=
and
(0) 3F=
, find
(4)F
.
Chapter 6
114.
The following figure is a graph of
‘( )fx
. On which of the following intervals is f
decreasing?
A)
–2 1x
B)
13x
C)
02x
D)
0.8 3x−  
Chapter 6
115.
The following figure is a graph of
‘( )fx
. Which of the following statements are
correct, assuming that the domain of f ‘ is [-2,3]? (Check all that apply.)
A)
-2 is a local maximum
B)
-2 is a local minimum
C)
1 is a local maximum
D)
1 is a local minimum
E)
3 is a local maximum
F)
3 is a local minimum
information on its derivative. difficulty: easy section: 6.7
Chapter 6
116.
Given the following graph of
‘( )gx
and the fact that
(0) 2000g=
, find
(100)g
.
Chapter 6
117.
Given the following graph of
‘( )gx
and the fact that
(0) 2000g=
, determine whether
(350)g
is positive or negative.
Chapter 6
118.
Given the following graph of
‘( )gx
and the fact that
(0) 2000g=
, what is x = 100?
A)
a local minimum
B)
an inflection point
C)
a local maximum
D)
none of the above
Chapter 6
119.
The following graph represents the rate of change of a function f with respect to x; i.e.,
it is the graph of
‘f
, with
(0) 0f=
. Which of the following are true at x = 1.8?
(Check all that apply.)
A)
f is concave up
B)
f is concave down
C)
f is increasing
D)
f is decreasing
Chapter 6
120.
The following graph represents the rate of change of a function f with respect to x; i.e.,
it is the graph of
‘f
, with
(0) 0f=
. Find a value a to one decimal place such that
02a
and
( ) 0fa=
. If there is no such value, enter “none”.
121.
The following figure shows the graph of
‘( )fx
. If
(0) 100f=
, find
(20)f
.
Chapter 6
122.
Given the values of
‘( )fx
in the table and that
(0) 40f=
, estimate
(2)f
to the
nearest whole number.
x
0
2
4
6
‘( )fx
3
15
27
39
123.
Using the following figure, find
(2)G
if
(0) –5G=
and
‘Gg=
.
124.
The following figure shows the graph of f. If
‘Ff=
and
(0) 0F=
, find
(2)F
.
Chapter 6
125.
True or False:
2
11
at at at
te dt te e C
aa
= − +
, where a is a constant.
A)
True
B)
False
difficulty: medium section: 6.7
126.
True or False:
ln lnx dx x x x C= − +
.
A)
True
B)
False
127.
True or False:
2 2 ( 1)
xx
xe dx e x C
−−
= + +
.
A)
True
B)
False
difficulty: medium section: 6.7
128.
Use integration by parts to find
7x
xe dx
.
difficulty: easy section: 6.7
129.
Use integration by parts to find
3
5x
xe dx
.
difficulty: easy section: 6.7
130.
Use integration by parts to find
8lnx xdx
.
difficulty: medium section: 6.7
Chapter 6
131.
Use integration by parts to find
5
2ln xdx
. (Give the exact answer in terms of natural
logs.)
132.
Use integration by parts to find
7
0cos x x dx
.