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(b)
ln x
x 2 ln x
(c)
ex
x 2 ln x
2 1
+ ’1 ·
(d)
ex
x ln x
x
2 1
’ ’1
(e)
x ln x
x
The derivative of f (x) = Sin’1 (x · ln x) is
68.
1 + ln x
(a)
x 2 · ln2 x
1
(b)
1 ’ x 2 · ln2 x
328 Final Exam

ln x
(c)
1 ’ x 2 · ln2 x
1 + ln x
(d)
1 ’ x 2 · ln2 x
1 + ln x

(e)
1 ’ x2
The value of the derivative of Tan’1 (ex · cos x) is
69.
ex
(a)
1 + e2x cos2 x
ex sin x
(b)
1 + e2x cos2 x
ex cos x
(c)
1 + e2x cos2 x


Y
ex (cos x ’ sin x)
FL
(d)
1 + cos2 x
ex (cos x ’ sin x)
AM

(e)
1 + e2x cos2 x
70. The value of the integral log3 x dx is
x
TE



(a) x ln x ’ +C
ln 3
(b) x log3 x ’ x + C
x
(c) x log3 x ’ +C
ln 3
x
(d) x log3 x ’ + C
3
x
(e) x ln x ’ + C
3
1 x2
· x dx is
71. The value of the integral 05
4
(a)
ln 5
ln 5
(b)
2
ln 5
(c)
ln 2
329
Final Exam

2
(d)
ln 5
ln 2
(e)
ln 5
x · 2x dx is
72. The value of the integral
x · 2x 2x
’ 2 +C
(a)
ln 2 ln 2
x·2 x 2x
+ 2 +C
(b)
ln 2 ln 2
x·2 x 2x
’ +C
(c)
ln 2 ln 2
2x
’ 2x + C
(d)
ln 2
2x
x
’ +C
(e)
ln 2 ln 2
73. A petri dish contains 7,000 bacteria at 10:00 a.m. and 10,000 bacteria at
1:00 p.m. How many bacteria will there be at 4:00 p.m.?
(a) 700000
(b) 10000
(c) 100000
(d) 10000/7
(e) 100000/7
74. There are 5 grams of a radioactive substance present at noon on January 1,
2005. At noon on January 1 of 2009 there are 3 grams present. When will
there be just 2 grams present?
= 5.127, or in early February of 2010
(a) t
= 7.712, or in mid-August of 2012
(b) t
= 7.175, or in early March of 2012
(c) t
= 6.135, or in early February of 2011
(d) t
= 6.712, or in mid-August of 2011
(e) t
75. If $8000 is placed in a savings account with 6% interest compounded
continuously, then how large is the account after ten years?
(a) 13331.46
(b) 11067.35
(c) 14771.05
(d) 13220.12
(e) 14576.95
330 Final Exam

76. A wealthy uncle wishes to ¬x an endowment for his favorite nephew.
He wants the fund to pay the young fellow $1,000,000 in cash on the
day of his thirtieth birthday. The endowment is set up on the day of the
nephew™s birth and is locked in at 8% interest compounded continuously.
How much principle should be put into the account to yield the necessary
payoff?
(a) 88,553.04
(b) 90,717.95
(c) 92,769.23
(d) 91,445.12
(e) 90,551.98

The values of Sin’1 1/2 and Tan’1 3 are
77.
(a) π/4 and π/3
(b) π/3 and π/2
(c) π/2 and π/3
(d) π/6 and π/3
(e) π/3 and π/6
dx
78. The value of the integral dx is
4 + x2
1 x
Tan’1 +C
(a)
2 2
1 x
Tan’1 +C
(b)
2 4
1 x
Tan’1 +C
(c)
4 2
x2
1
Tan’1 +C
(d)
2 2
1 2
Tan’1 +C
(e)
2 x
ex dx

79. The value of the integral dx
1 ’ e2x
Cos’1 ex + C
(a)
Sin’1 e2x + C
(b)
Sin’1 e’x + C
(c)
Cos’1 e’2x + C
(d)
Sin’1 ex + C
(e)
331
Final Exam

2 x dx

80. The value of the integral dx is
2 x4 ’ 1
x
1
π
(a)
3
’π
(b)
4
’π
(c)
6
π
(d)
4
π
(e)
6
x 2 ln x dx is
81. The value of the integral

x3 x3
ln x ’ +C
(a)
3 9
x2 x2
ln x ’ +C
(b)
3 9
x3 x3
ln x ’ +C
(c)
2 6
x3 x3
ln x ’ +C
(d)
9 3
x5 x3
ln x ’ +C
(e)
3 6
1x
82. The value of the integral sin x dx is
0e
e · cos 1 ’ e · sin 1 + 1
(a)
e · sin 1 ’ e · cos 1 ’ 1
(b)
e · sin 1 ’ e · cos 1 + 1
(c)
e · sin 2 ’ e · cos 2 + 1
(d)
e · sin 2 + e · cos 2 ’ 1
(e)
x · e2x dx is
83. The value of the integral
xex ex
’ +C
(a)
2 4
xe2x e2x
’ +C
(b)
4 2
xex ex
’ +C
(c)
4 2
332 Final Exam

xe2x ex
’ +C
(d)
2 4
xe2x e2x
’ +C
(e)
2 4
dx
84. The value of the integral is
x(x + 1)
ln |x + 1| ’ ln |x| + C
(a)
ln |x ’ 1| ’ ln |x + 1| + C
(b)
ln |x| ’ ln |x + 1| + C
(c)
ln |x| ’ ln |x| + C
(d)
ln |x + 2| ’ ln |x + 1| + C
(e)
dx
85. The value of the integral is
x(x 2 + 4)
1 1
ln |x| ’ ln(x 2 + 2) + C
(a)
2 8
1 1
ln |x| ’ ln(x 2 + 4) + C
(b)
4 8
1 1
ln |x| ’ ln(x 2 + 4) + C
(c)
8 4
1 1
ln |x| ’ ln(x 2 + 1) + C
(d)
2 8
1 1
ln |x| ’ ln(x 2 + 4) + C
(e)
8 2
dx
86. The value of the integral is
(x ’ 1)2 (x + 1)
1 1 1
ln |x ’ 1| + ’ ln |x + 1| + C
(a)
x’1 2
2
’1 1 1
ln |x ’ 1| ’ + ln |x + 1| + C
(b)
(x ’ 1)2 2
2
’1 1 1
ln |x ’ 1| + + ln |x + 1| + C
(c)
x’1 4
2
’1 1 1
ln |x ’ 1| ’ + ln |x + 1| + C
(d)
x’1 2
2
’1 1/2 1
ln |x ’ 1| ’ + ln |x + 1| + C
(e)
x’1 4
4
2√
The value of the integral 1 x 1 + x 2 dx is
87.
333
Final Exam

’ 1 23/2
1 3/2
(a) 45 4
’ 1 33/2
1 3/2
(b) 34 3
’ 1 23/2
1 3/2
(c) 35 3
’ 1 33/2
1 3/2
(d) 54 5
’ 1 53/2
1 3/2
(e) 32 3
π/4 sin x cos x
88. The value of the integral dx
1 + cos2 x
0

3
(a) ln
5

5
(b) ln
2√
1 3
(c) ln
2√2
2
(d) ln
3
2
ln √
(e)
3
1x
sin(1 + ex ) dx is
89. The value of the integral 0e
’ sin(1 + e) + cos 2
(a)
’ sin(1 ’ e) + sin 2
(b)
’ cos(1 + e) + cos 2
(c)
cos(1 + e) ’ cos 2
(d)

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