Solve the problem. Round your answer, if appropriate.
99)
The following table shows the rate of water flow (in gal/min) from a stream into a pond during a
30–minute period after a thunderstorm. Use Simpson’s rule to estimate the total amount of water
flowing into the pond during this period.
Time (min) Rate (gal/min)
0200
5250
10 300
15 250
20 220
25 200
30 150
99)
A)
6975 gallons
B)
6383.3 gallons
C)
7975 gallons
D)
6983.3 gallons
Evaluate the integral.
100)
2
0
24x
(4x2+3)2 dx
100)
A)
16
171
B)
16
19
C)
10
19
D)
6832
10,556,001
Determine the integral by making an appropriate substitution.
101)
12x36x2–13 dx
101)
A)
(6x2+13)4
24 + C
B)
(6x2–13)4/3
3+ C
C)
3(6x2+13)4/3
4+ C
D)
3(6x2–13)4/3
4+ C
Evaluate the integral.
102)
4
1
1
2x + 1 dx
102)
A)
ln 3
2
B)
–ln 2
C)
1 + ln 2
D)
ln 4
Evaluate the integral using integration by parts.
103)
sin–1 x dx
103)
A)
sin–1 x +1 –x2+ C
B)
sin–1 x –1 –x2+ C
C)
x +1 –x2+ C
D)
x sin–1 x +1 –x2+ C
Determine the integral by making an appropriate substitution.
104)
sin5 x cos x dx
104)
A)
sin6x
6+ C
B)
sin5x
6+ C
C)
sin6x
5+ C
D)
sin5x
5+ C
105)
(x – 1)7 dx
105)
A)
1
16(x – 1)8+ C
B)
1
9(x – 1)9+1
8(x – 1)8+ C
C)
1
2x2+1
8(x – 1)8+ C
D)
1
8(x – 1)8+ C
Evaluate the integral using integration by parts.
106)
ln 2x dx
106)
A)
x(ln|2x| + 2) + C
B)
x(ln|2x| – 1) + C
C)
x(ln|2x| – 2) + C
D)
x(ln|2x| + 1) + C
107)
eax cos bx dx
107)
A)
eax(sin bx – cos bx) + C
B)
1
2b eax cos2 bx + C
C)
eax(sin bx + cos bx) + C
D)
eax b sin bx + a cos bx
a2+b2+ C
Determine the integral by making an appropriate substitution.
108)
sin x
1 – cos x dx
108)
A)
–ln 1 – cos x + C
B)
ln 1 – cos x + C
C)
sin x cos x + C
D)
1
2(1 – cos x)2+ C
E)
none of these
Evaluate the integral using integration by parts.
109)
xe8x dx
109)
A)
e8x 1
8x –1
64 + C
B)
e8x x –1
8+ C
C)
e8x(8x – 1) + C
D)
xe8x + C
Determine the integral by making an appropriate substitution.
110)
1
6 x
53+x dx
110)
A)
(3 +x)6
18 + C
B)
5(3 +x)6/5
36 + C
C)
5(3 +x)6/5
6+ C
D)
5(3 +x)6/5
18 + C
Solve the problem. Round your answer, if appropriate.
111)
Suppose that the accompanying table shows the velocity of a car every second for 8 seconds. Use
the trapezoidal rule to approximate the distance traveled by the car in the 8 seconds.
Time (sec) Velocity (ft/sec)
015
116
217
319
418
520
617
715
816
111)
A)
153 feet
B)
209.5 feet
C)
137.5 feet
D)
275 feet
Evaluate the integral using integration by parts.
112)
x 5 – x dx
112)
A)
–2
3x(5 – x)3/2 –2
5(5 – x)5/2 + C
B)
2
3x(5 – x)3/2 +4
15 (5 – x)5/2 + C
C)
–2
3x(5 – x)3/2 –4
15 (5 – x)5/2 + C
D)
–2
3x(5 – x)3/2 +4
15 (5 – x)5/2 + C
Determine the integral by making an appropriate substitution.
113)
x6
ex7 dx
113)
A)
–1
7ex7+ C
B)
–1
7ex7–1
+ C
C)
6x5
ex7+ C
D)
1
ex7+ C
Evaluate the integral using integration by parts.
114)
sec4 x dx
114)
A)
4 sec3 x tan x+ C
B)
sec2 x tan x + tan x + C
C)
1
3sec2 x tan x +2
3tan x + C
D)
1
5sec5 x tan x + C
Determine the integral by making an appropriate substitution.
115)
9x2–2
(3x3–2x)2 dx
115)
A)
1
3x3–2x
+ C
B)
(3x3–2x)3
3+ C
C)
–1
3x3–2x
+ C
D)
(3x3–2x)2
2+ C
Solve the problem.
116)
The rate of a continuous money flow is 500e0.04 dollars per year for 10 years. Find the present
value if interest is earned at 6% compounded continuously.
116)
A)
$45,468.27
B)
$5535.07
C)
$30,535.07
D)
$4531.73
Determine the integral by making an appropriate substitution.
117)
x5x6+5 dx
117)
A)
–1
3(x6+5)–1/2 + C
B)
2
3(x6+5)3/2 + C
C)
4(x6+5)3/2 + C
D)
1
9(x6+5)3/2 + C
118)
Find the area of the shaded region.
y =6x
(3x2+ 1)2
118)
A)
– 1 square units
B)
3
7 square units
C)
2 square units
D)
11
7 square units
Evaluate the improper integral whenever it is convergent. If it is divergent, state this.
119)
2
dx
x ln x
119)
A)
2
ln 2
B)
ln 2
C)
2 ln 2
D)
divergent
E)
ln 2
2
Determine the integral by making an appropriate substitution.
120)
x +3
2
x2+ 3x dx
120)
A)
1
2 ln x2+ 3x + C
B)
(x2+ 3x)2+ C
C)
(x2+ 3x)–1+ C
D)
ln x2+ 3x + C
E)
none of these
121)
tan9(2x) sec2(2x) dx
121)
A)
9 tan8 x + C
B)
1
20 tan10 x + C
C)
1
10 tan10 x + C
D)
tan10 x + C
E)
none of these
122)
7x245+2x3 dx
122)
A)
7(5 +2x3)5/4 + C
B)
–14
3(5 +2x3)–3/4 + C
C)
14
15 (5 +2x3)5/4 + C
D)
28
5(5 +2x3)5/4 + C
Evaluate the improper integral whenever it is convergent. If it is divergent, state this.
123)
1
dx
x2
123)
A)
divergent
B)
1
C)
1
4
D)
0
E)
1
2
Determine the integral by making an appropriate substitution.
124)
x2sec2x3dx
124)
A)
x2 sec x3+ C
B)
tan2 x3+ C
C)
tan x3+ C
D)
1
3 tan x3+ C
E)
none of these
Evaluate the improper integral whenever it is convergent. If it is divergent, state this.
125)
0
3
(x + 1)2 dx
125)
A)
3
B)
0
C)
–3
D)
Divergent
Determine the integral by making an appropriate substitution.
126)
9x
x –5 dx
126)
A)
6x +5(x –10) + C
B)
27
2(x –5)3/2 + C
C)
6x –5(x +10) + C
D)
6(x –5)3/2 + C
127)
ln x
xdx Use the substitution u =ln x.
127)
A)
2(ln x)2+ C
B)
1
2(ln x)2+ C
C)
(ln x)2+ C
D)
(ln x)2
4+ C
E)
none of these
128)
If the annual rate of return from an investment is –3000 + 125t, find the present value of the income
generated in the third year if the interest rates are 8.5%.
128)
A)
3
0
(125t – 3000)e0.085t dt
B)
3
2
(–3000 + 125t)e–0.085t dt
C)
3
2
(125t – 3000)e0.085t dt
D)
3
0
(–3000 + 125t)e–0.085t dt
E)
none of these
129)
Suppose the annual rate of income from an investment at any time t is K(t) = – 100 + 50t. What is the
formula for the present value of the income over the next 5 years at a 6% interest rate compounded
continuously?
129)
A)
5
0
(50t – 100)e–0.06t dt
B)
5
1
(50t – 100)e–0.06t dt
C)
5
0
(50t –100)e0.06t dt
D)
5
1
50t–0.06t dt
E)
none of these
30
Evaluate the integral using integration by parts.
130)
16x sin x dx
130)
A)
16 sin x – 16 cos x + C
B)
16 sin x + 16x cos x + C
C)
16 sin x – x cos x + C
D)
16 sin x – 16x cos x + C
Evaluate the improper integral whenever it is convergent. If it is divergent, state this.
131)
0
e10x dx
131)
A)
divergent
B)
1
C)
–1
D)
0
E)
1
10
Determine the integral by making an appropriate substitution.
132)
tan2 x sec2 x dx Use the substitution u = tan x.
132)
A)
tan2 x
2+ C
B)
sec2 x
2+ C
C)
tan3 x
3+ C
D)
sec x tan x + C
E)
none of these
Solve the problem. Round your answer, if appropriate.
133)
The following table shows the rate of water flow (in gal/min) from a stream into a pond during a
30–minute period after a thunderstorm. Use the trapezoidal rule to estimate the total amount of
water flowing into the pond during this period.
Time (min) Rate (gal/min)
0200
5250
10 300
15 250
20 220
25 200
30 150
133)
A)
6383.3 gallons
B)
7850 gallons
C)
6983.3 gallons
D)
6975 gallons
134)
Suppose that the accompanying table shows the velocity of a car every second for 8 seconds. Use
Simpson’s rule to approximate the distance traveled by the car in the 8 seconds.
Time (sec) Velocity (ft/sec)
019
120
221
323
422
524
621
719
820
134)
A)
126.00 feet
B)
169.50 feet
C)
168.33 feet
D)
170.33 feet
Evaluate the integral using integration by parts.
135)
xx + 1 dx
135)
A)
x(x + 1)3/2 –(x + 1)5/2 + C
B)
2
3(x + 1)3/2 –4
15 (x + 1)5/2 + C
C)
(x + 1)3/2 +(x + 1)5/2 + C
D)
2
3x(x + 1)3/2 –4
15 (x + 1)5/2 + C
136)
ln 3x
x7 dx
136)
A)
–1
6 x–6 ln 3x +1
36 x–6+ C
B)
ln 3x +1
6 x–6+ C
C)
–1
6 x–6 ln 3x –1
30 x–5+ C
D)
–1
6 x–6 ln 3x –1
36 x–6+ C
Evaluate the integral.
137)
1
0
x2
(2 +5x3)2 dx
137)
A)
1
42
B)
75
14
C)
3
2
D)
67
9
Determine the integral by making an appropriate substitution.
138)
10e3x dx
138)
A)
10
3e3x + C
B)
10e3x+1
3x + 1 + C
C)
10e3x + C
D)
1
3e3x + C