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Appendix G.1
G
■
Differentiation and Integration Formulas
Formulas
G.1 Differentiation and Integration Formulas
■
Use differentiation and integration tables to supplement differentiation and integration techniques.
Differentiation Formulas
1.
d
关cu兴 ⫽ cu⬘
dx
2.
d
关u ± v兴 ⫽ u⬘ ± v⬘
dx
3.
d
关uv兴 ⫽ uv⬘ ⫹ vu⬘
dx
4.
d u
vu⬘ ⫺ uv⬘
⫽
dx v
v2
冤冥
5.
d
关c兴 ⫽ 0
dx
6.
d n
关u 兴 ⫽ nun⫺1u⬘
dx
7.
d
关x兴 ⫽ 1
dx
8.
d
u⬘
关ln u兴 ⫽
dx
u
9.
d u
关e 兴 ⫽ e uu⬘
dx
10.
d
关 sin u兴 ⫽ 共cos u兲u⬘
dx
11.
d
关cos u兴 ⫽ ⫺ 共sin u兲u⬘
dx
12.
d
关tan u兴 ⫽ 共sec2 u兲u⬘
dx
13.
d
关cot u兴 ⫽ ⫺ 共csc2 u兲u⬘
dx
14.
d
关sec u兴 ⫽ 共sec u tan u兲u⬘
dx
15.
d
关csc u兴 ⫽ ⫺ 共csc u cot u兲u⬘
dx
Integration Formulas
Forms Involving u n
1.
冕
un du ⫽
un⫹1
⫹ C, n ⫽ ⫺1
n⫹1
2.
冕
1
du ⫽ ln u ⫹ C
u
ⱍⱍ
Forms Involving a ⴙ bu
3.
4.
5.
6.
7.
8.
9.
10.
冕
冕
冕
冕
冕
冕
冕
冕
u
1
du ⫽ 2共bu ⫺ a ln a ⫹ bu 兲 ⫹ C
a ⫹ bu
b
ⱍ
ⱍ
冢
冣
u
1
a
du ⫽ 2
⫹ lnⱍa ⫹ buⱍ ⫹ C
共a ⫹ bu兲2
b a ⫹ bu
u
1
⫺1
a
du ⫽ 2
⫹
⫹ C, n ⫽ 1, 2
共a ⫹ bu兲n
b 共n ⫺ 2兲共a ⫹ bu兲n⫺2 共n ⫺ 1兲共a ⫹ bu兲n⫺1
冤
冥
冤
ⱍ冥 ⫹ C
u2
1
bu
du ⫽ 3 ⫺ 共2a ⫺ bu兲 ⫹ a2 ln a ⫹ bu
a ⫹ bu
b
2
ⱍ
冢
ⱍ冣 ⫹ C
u2
1
a2
du
⫽
bu
⫺
⫺ 2a ln a ⫹ bu
共a ⫹ bu兲2
b3
a ⫹ bu
ⱍ
冤
ⱍ冥 ⫹ C
u2
1
2a
a2
du ⫽ 3
⫺
⫹ ln a ⫹ bu
3
共a ⫹ bu兲
b a ⫹ bu 2共a ⫹ bu兲2
ⱍ
u2
1
⫺1
2a
a2
du ⫽ 3
⫹
⫺
⫹ C,
n
n⫺3
n⫺2
共a ⫹ bu兲
b 共n ⫺ 3兲共a ⫹ bu兲
共n ⫺ 2兲共a ⫹ bu兲
共n ⫺ 1兲共a ⫹ bu兲n⫺1
冤
ⱍ ⱍ
1
1
u
du ⫽ ln
⫹C
u共a ⫹ bu兲
a a ⫹ bu
冥
n ⫽ 1, 2, 3
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Appendix G
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1:47 PM
Formulas
Integration Formulas
11.
12.
13.
冕
冕
冕
Page G2
(continued)
ⱍ ⱍ冣
ⱍ ⱍ冣
ⱍ ⱍ冥
冢
1
1
1
1
u
du ⫽
⫹ ln
u共a ⫹ bu兲2
a a ⫹ bu a a ⫹ bu
冢
1
1 1 b
u
du ⫽ ⫺
⫹ ln
u2共a ⫹ bu兲
a u a a ⫹ bu
⫹C
⫹C
1
1 a ⫹ 2bu
2b
u
du ⫽ ⫺ 2
⫹
ln
u2共a ⫹ bu兲2
a u共a ⫹ bu兲
a
a ⫹ bu
冤
Forms Involving 冪a ⴙ bu
14.
15.
16.
17.
18.
19.
20.
冕
冕
冕
冕
冕
冕
冕
un 冪a ⫹ bu du ⫽
冤
ⱍ
ⱍ
冪a ⫹ bu ⫺ 冪a
1
1
du ⫽
ln
⫹ C,
冪a
冪a ⫹ bu ⫹ 冪a
u冪a ⫹ bu
冤
冪a ⫹ bu
u
冪a ⫹ bu
un
u
冪a ⫹ bu
冕
du ⫽
22.
冕
冕
冕
1
du
u冪a ⫹ bu
du ⫽ 2冪a ⫹ bu ⫹ a
⫺1
共a ⫹ bu兲3兾2 共2n ⫺ 5兲b
⫹
a共n ⫺ 1兲
un⫺1
2
冤
du ⫽ ⫺
2共2a ⫺ bu兲
冪a ⫹ bu ⫹ C
3b2
冢
un
2
du ⫽
un冪a ⫹ bu ⫺ na
共2n ⫹ 1兲b
冪a ⫹ bu
1
du ⫽ ⫺
u2 ⫺ a2
冕
冕
冥
un⫺1冪a ⫹ bu du
a > 0
冪a ⫹ bu
1
⫺1
共2n ⫺ 3兲b
du ⫽
⫹
a共n ⫺ 1兲
un⫺1
2
un冪a ⫹ bu
Forms Involving u 2 ⴚ a 2, a > 0
21.
冕
2
un共a ⫹ bu兲3兾2 ⫺ na
b共2n ⫹ 3兲
⫹C
冥
1
du , n ⫽ 1
un⫺1冪a ⫹ bu
冕
冪a ⫹ bu
un⫺1
un⫺1
du
冪a ⫹ bu
ⱍ ⱍ
u⫺a
1
1
du ⫽
ln
⫹C
a2 ⫺ u2
2a u ⫹ a
冕
1
⫺1
u
du ⫽ 2
⫹ 共2n ⫺ 3兲
共u2 ⫺ a2兲n
2a 共n ⫺ 1兲 共u2 ⫺ a2兲n⫺1
冤
冥
du , n ⫽ 1
冣
冥
1
du , n ⫽ 1
共u2 ⫺ a2兲n⫺1
Forms Involving 冪u 2 ± a 2, a > 0
23.
24.
25.
26.
27.
28.
冕
冕
冕
冕
冕
冕
冪u2 ± a2 du ⫽
1
共u冪u2 ± a2 ± a2 ln u ⫹ 冪u2 ± a2 兲 ⫹ C
2
ⱍ
u2冪u2 ± a2 du ⫽
冪u2 ⫹ a2
u
冪u2 ± a2
u2
1
冪u2 ± a2
ⱍ
u
⫹
ⱍ
a2
ⱍ
a ⫹ 冪u2 ⫹ a2
⫹C
u
⫺ 冪u2 ± a2
⫹ ln u ⫹ 冪u2 ± a2 ⫹ C
u
ⱍ
ⱍ
ⱍ
ⱍ
du ⫽ ln u ⫹ 冪u2 ± a2 ⫹ C
1
冪u2
1
关u共2u2 ± a2兲冪u2 ± a2 ⫺ a4 ln u ⫹ 冪u2 ± a2 兴 ⫹ C
8
du ⫽ 冪u2 ⫹ a2 ⫺ a ln
du ⫽
ⱍ
du ⫽
ⱍ
ⱍ
⫺1 a ⫹ 冪u2 ⫹ a2
ln
⫹C
a
u
ⱍ
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Appendix G.1
29.
30.
31.
冕
冕
冕
u2
冪u ± a
2
2
du ⫽
1
共u冪u2 ± a2 ⫿ a2 ln u ⫹ 冪u2 ± a2 兲 ⫹ C
2
ⱍ
33.
34.
冕
冕
冕
ⱍ
冪u2 ± a2
1
du
⫽
⫹C
⫿
a2u
u2冪u2 ± a2
1
±u
du ⫽ 2 2
⫹C
共u2 ± a2兲3兾2
a 冪u ± a2
Forms Involving 冪a 2 ⴚ u 2, a > 0
32.
冪a2 ⫺ u2
u
1
u冪a ⫺ u
2
ⱍ
du ⫽ 冪a2 ⫺ u2 ⫺ a ln
2
du ⫽
ⱍ
ⱍ
a ⫹ 冪a2 ⫺ u2
⫹C
u
⫺1 a ⫹ 冪a ⫺ u
ln
a
u
2
1
⫺ 冪a2 ⫺ u2
du
⫽
⫹C
a2u
u2冪a2 ⫺ u2
2
ⱍ
⫹C
35.
Forms Involving e u
36.
38.
40.
冕
冕
冕
eu du ⫽ eu ⫹ C
37.
冕
uneu du ⫽ uneu ⫺ n
un⫺1eu du
39.
43.
44.
冕
冕
冕
ln u du ⫽ u共⫺1 ⫹ ln u兲 ⫹ C
un ln u du ⫽
42.
un⫹1
关⫺1 ⫹ 共n ⫹ 1兲 ln u兴 ⫹ C, n ⫽ ⫺1
共n ⫹ 1兲2
共ln u兲2 du ⫽ u关2 ⫺ 2 ln u ⫹ 共ln u兲2兴 ⫹ C
45.
Forms Involving sin u or cos u
46.
48.
50.
51.
52.
54.
冕
冕
冕
冕
冕
冕
冕
冕
冕
1
u
du ⫽ 2 2
⫹C
共a2 ⫺ u2兲3兾2
a 冪a ⫺ u2
ueu du ⫽ 共u ⫺ 1兲eu ⫹ C
1
du ⫽ u ⫺ ln共1 ⫹ eu兲 ⫹ C
1 ⫹ eu
1
1
du ⫽ u ⫺ ln共1 ⫹ enu兲 ⫹ C
1 ⫹ enu
n
Forms Involving ln u
41.
Differentiation and Integration Formulas
■
sin u du ⫽ ⫺cos u ⫹ C
47.
1
sin2 u du ⫽ 共u ⫺ sin u cos u兲 ⫹ C
2
49.
sinn u du ⫽ ⫺
cosn u du ⫽
冕
冕
sinn⫺1 u cos u n ⫺ 1
⫹
n
n
cosn⫺1 u sin u n ⫺ 1
⫹
n
n
冕
u ln u du ⫽
冕
冕
冕
u2
共⫺1 ⫹ 2 ln u兲 ⫹ C
4
共ln u兲n du ⫽ u共ln u兲n ⫺ n
冕
共ln u兲n⫺1 du
cos u du ⫽ sin u ⫹ C
1
cos2 u du ⫽ 共u ⫹ sin u cos u兲 ⫹ C
2
sinn⫺2 u du
cosn⫺2 u du
u sin u du ⫽ sin u ⫺ u cos u ⫹ C
un sin u du ⫽ ⫺un cos u ⫹ n
冕
un⫺1 cos u du
53.
冕
u cos u du ⫽ cos u ⫹ u sin u ⫹ C
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Appendix G
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Formulas
Integration Formulas
55.
56.
57.
58.
冕
冕
冕
冕
Page G4
(continued)
冕
un cos u du ⫽ un sin u ⫺ n
un⫺1 sin u du
1
du ⫽ tan u ⫿ sec u ⫹ C
1 ± sin u
1
du ⫽ ⫺cot u ± csc u ⫹ C
1 ± cos u
1
du ⫽ ln tan u ⫹ C
sin u cos u
ⱍ
ⱍ
Forms Involving tan u, cot u, sec u, or csc u
59.
60.
61.
62.
63.
64.
65.
66.
67.
68.
69.
70.
71.
72.
73.
74.
冕
冕
冕
冕
冕
冕
冕
冕
冕
冕
冕
冕
冕
冕
冕
冕
ⱍ
ⱍ
tan u du ⫽ ⫺ln cos u ⫹ C
ⱍ
ⱍ
cot u du ⫽ ln sin u ⫹ C
ⱍ
ⱍ
ⱍ
ⱍ
sec u du ⫽ ln sec u ⫹ tan u ⫹ C
csc u du ⫽ ln csc u ⫺ cot u ⫹ C
tan2 u du ⫽ ⫺u ⫹ tan u ⫹ C
cot2 u du ⫽ ⫺u ⫺ cot u ⫹ C
sec2 u du ⫽ tan u ⫹ C
csc2 u du ⫽ ⫺cot u ⫹ C
tann u du ⫽
tann⫺1 u
⫺
n⫺1
cotn u du ⫽ ⫺
secn u du ⫽
冕
cotn⫺1 u
⫺
n⫺1
冕
cotn⫺2 u du,
secn⫺2 u tan u n ⫺ 2
⫹
n⫺1
n⫺1
cscn u du ⫽ ⫺
n⫽1
tann⫺2 u du,
冕
cscn⫺2 u cot u n ⫺ 2
⫹
n⫺1
n⫺1
n⫽1
secn⫺2 u du,
冕
cscn⫺2 u du, n ⫽ 1
1
1
du ⫽ 共u ± ln cos u ± sin u 兲 ⫹ C
1 ± tan u
2
ⱍ
ⱍ
1
1
du ⫽ 共u ⫿ ln sin u ± cos u 兲 ⫹ C
1 ± cot u
2
ⱍ
ⱍ
1
du ⫽ u ⫹ cot u ⫿ csc u ⫹ C
1 ± sec u
1
du ⫽ u ⫺ tan u ± sec u ⫹ C
1 ± csc u
n⫽1
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Appendix G.2
■
Formulas from Business and Finance
G.2 Formulas from Business and Finance
■
Summary of business and finance formulas
Formulas from Business
Basic Terms
x ⫽ number of units produced (or sold)
p ⫽ price per unit
R ⫽ total revenue from selling x units
C ⫽ total cost of producing x units
C ⫽ average cost per unit
P ⫽ total profit from selling x units
Basic Equations
R ⫽ xp
C⫽
C
x
P⫽R⫺C
Typical Graphs of Supply and Demand Curves
p
Supply curves increase as price
increases and demand curves
decrease as price increases. The
equilibrium point occurs when the
supply and demand curves intersect.
Demand
Equilibrium
p0
price
Supply
Equilibrium point
(x0, p0)
x
x0
Equilibrium quantity
Demand Function: p ⴝ f 冇x冈 ⴝ price required to sell x units
␩⫽
p兾x
⫽ price elasticity of demand
dp兾dx
共When ⱍ␩ⱍ < 1, the demand is inelastic. When ⱍ␩ⱍ > 1, the demand is elastic.兲
Typical Graphs of Revenue, Cost, and Profit Functions
R
C
Elastic
demand
P
Inelastic
demand
Maximum
profit
x
Fixed
cost
Break-even
point
x
x
Negative of
fixed cost
Revenue Function
The low prices required to
sell more units eventually
result in a decreasing
revenue.
Cost Function
The total cost to produce
x units includes the fixed
cost.
Profit Function
The break-even point occurs
when R ⫽ C.
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Formulas
Formulas from Business
(continued)
Marginals
dR
⫽ marginal revenue ⬇ the extra revenue from selling one additional unit
dx
dC
⫽ marginal cost ⬇ the extra cost of producing one additional unit
dx
dP
⫽ marginal profit ⬇ the extra profit from selling one additional unit
dx
Marginal
revenue
1 unit
Extra revenue
for one unit
Revenue Function
Formulas from Finance
Basic Terms
P ⫽ amount of deposit
n ⫽ number of times interest is compounded per year
t ⫽ number of years
r ⫽ interest rate
A ⫽ balance after t years
Compound Interest Formulas
冢
1. Balance when interest is compounded n times per year: A ⫽ P 1 ⫹
r
n
冣
nt
2. Balance when interest is compounded continuously: A ⫽ Pert
Effective Rate of Interest
冢
reff ⫽ 1 ⫹
r
n
冣
n
⫺1
Present Value of a Future Investment
A
P⫽
冢1 ⫹ nr 冣
nt
Balance of an Increasing Annuity After n Deposits of P per Year for t Years
冤 冢1 ⫹ nr 冣
A⫽P
nt
冥冢
⫺1 1⫹
n
r
冣
Initial Deposit for a Decreasing Annuity with n Withdrawals of W per Year for t Years
P⫽W
冢nr冣冦1 ⫺ 冤 1 ⫹ 1共r兾n兲冥 冧
nt
Monthly Installment M for a Loan of P Dollars over t Years at r% Interest
冦
M⫽P
r兾12
1
1⫺
1 ⫹ 共r兾12兲
冤
冥
12t
冧
Amount of an Annuity
冕
T
erT
c共t兲e⫺rt dt
0
c共t兲 is the continuous income function in dollars per year and T is the term of the annuity in years.