FORMULAS

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APPENDIX B
B
B.1
Page A12
Formulas
FORMULAS
D I F F E R E N T I AT I O N A N D I N T E G R AT I O N F O R M U L A S
Use differentiation and integration tables to supplement differentiation and integration techniques.
Differentiation Formulas
1.
d
cu
cu
dx
2.
d
u ± v
u ± v
dx
3.
d
uv
uv vu
dx
4.
d u
vu uv
dx v
v2
5.
d
c
0
dx
6.
d n
u nun1u
dx
7.
d
x
1
dx
8.
u
d
ln u
dx
u
9.
d u
e e uu
dx
10.
d
sin u
cos uu
dx
11.
d
cos u
sin uu
dx
12.
d
tan u
sec2 uu
dx
13.
d
cot u
csc2 uu
dx
14.
d
sec u
sec u tan uu
dx
15.
d
csc u
csc u cot uu
dx
Integration Formulas
Forms Involving u n
1.
2.
un du un1
C, n 1
n1
1
du ln u C
u
Forms Involving a bu
3.
4.
5.
6.
7.
u
1
du 2bu a ln a bu C
a bu
b
u
1
a
du 2
lna bu C
a bu2
b a bu
u
1
1
a
du 2
C, n 1, 2
a bun
b n 2a bun2 n 1a bun1
u2
1
bu
du 3 2a bu a2 ln a bu
a bu
b
2
C
C
u2
1
a2
du 3 bu 2a ln a bu
2
a bu
b
a bu
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APPENDIX B
8.
9.
11.
12.
13.
u2
1
1
2a
du 3
a bun
b n 3a bun3 n 2a bun2
10.
15.
16.
17.
18.
19.
20.
a2
C,
n 1a bun1
n 1, 2, 3
1
1
u
du ln
C
ua bu
a a bu
1
1
1
1
u
du ln
ua bu2
a a bu a a bu
1
1 1 b
u
du ln
a bu
a u a a bu
u2
C
C
1
1 a 2bu
2b
u
du 2
ln
u2a bu2
a ua bu
a
a bu
Forms Involving a bu
14.
C
u2
1
2a
a2
du
ln a bu
a bu3
b3 a bu 2a bu2
un a bu du 2
una bu32 na
b2n 3
C
un1a bu du
a bu a
1
1
du ln
C, a > 0
a
a bu a
ua bu
a bu
1
1
2n 3b
du an 1
un1
2
una bu
a bu
u
a bu
un
u
a bu
du 2a bu a
du 1
du
ua bu
1
a bu32 2n 5b
an 1
un1
2
du 22a bu
a bu C
3b2
un
2
du una bu na
2n
1b
a bu
1
du ,
un1a bu
a bu
un1
un1
du
a bu
du ,
n1
n1
Forms Involving u 2 a 2, a > 0
21.
1
du u2 a2
22.
u2
1
du
a2 u2
ua
1
ln
C
2a u a
u
1
1
du 2
2n 3
2
n
2
a
2a n 1 u a2n1
u2
1
du ,
a2n1
n1
Formulas
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Formulas
APPENDIX B
Integration Formulas
(Continued)
Forms Involving u2 ± a2, a > 0
23.
24.
25.
26.
27.
28.
29.
30.
31.
u2 ± a2 du 1
uu2 ± a2 ± a2 ln u u2 ± a2 C
2
1
u2u2 ± a2 du u2u2 ± a2u2 ± a2 a4 ln u u2 ± a2 C
8
u2 a2
u
u2
±
a2
du u2
1
du 2
uu2 a
u
±
a2
u
1
u2 ± a2
u2
33.
34.
35.
du a u2 a2
C
u
ln u u2 ± a2 C
1 a u2 a2
ln
C
a
u
1
uu2 ± a2 a2 ln u u2 ± a2 C
2
u2 ± a2
1
du C
2
2
a2u
u u ± a
2
1
±u
du 2 2
C
u2 ± a232
a u ± a2
a2 u2
u
du a2 u2 a ln
1
1
du ln
a
ua2 u2
a a2 u2
u
1
u
du 2 2
C
a2 u232
a a u2
eu du eu C
a a2 u2
C
u
1
a2 u2
du
C
a2u
u2a2 u2
Forms Involving e u
36.
du ln u u2 ± a2 C
u2 ± a2
2
du u2 a2 a ln
Forms Involving a 2 u2, a > 0
32.
C
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APPENDIX B
37.
38.
39.
40.
ueu du u 1eu C
uneu du uneu n
un1eu du
1
du u ln1 eu C
1 eu
1
1
du u ln1 enu C
1 enu
n
Forms Involving In u
41.
42.
43.
44.
45.
ln u du u1 ln u C
u ln u du u2
1 2 ln u C
4
un ln u du un1
1 n 1 ln u
C,
n 12
n 1
ln u2 du u2 2 ln u ln u2
C
ln un du uln un n
ln un1 du
Forms Involving sin u or cos u
46.
47.
48.
49.
50.
51.
52.
sin u du cos u C
cos u du sin u C
1
sin2 u du u sin u cos u C
2
1
cos2 u du u sin u cos u C
2
sinn u du cosn u du sinn1 u cos u n 1
n
n
cosn1 u sin u n 1
n
n
u sin u du sin u u cos u C
sinn2 u du
cosn2 u du
Formulas
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Formulas
APPENDIX B
Integration Formulas
53.
54.
55.
56.
57.
58.
(Continued)
u cos u du cos u u sin u C
un sin u du un cos u n
un cos u du un sin u n
un1 cos u du
un1 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
Forms Involving tan u, cot u, sec u, or csc u
59.
60.
61.
62.
63.
64.
65.
66.
67.
68.
69.
70.
tan u du ln cos u C
cot u du ln sin u C
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 tann1 u
n1
cotn u du secn u du cotn1 u
n1
cotn2 u du,
secn2 u tan u n 2
n1
n1
cscn u du n1
tann2 u du,
cscn2 u cot u n 2
n1
n1
n1
secn2 u du, n 1
cscn2 u du,
n1
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APPENDIX B
71.
72.
73.
74.
B.2
1
1
du u ± ln cos u ± sin u C
1 ± tan u
2
1
1
du u ln sin u ± cos u C
1 ± cot u
2
1
du u cot u csc u C
1 ± sec u
1
du u tan u ± sec u C
1 ± csc u
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
PRC
Typical Graphs of Supply and Demand Curves
p
Demand
Equilibrium
p0
price
Supply
Equilibrium point
(x0, p0 )
x
x0
Equilibrium quantity
Supply curves increase as price increases and demand curves decrease as price
increases. The equilibrium point occurs when the supply and demand curves intersect.
Formulas
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APPENDIX B
Page A18
Formulas
Formulas from Business
(Continued)
Demand Function: p f x price required to sell x units
px
price elasticity of demand
dpdx
If < 1, the demand is inelastic. If > 1, the demand is elastic.
Typical Graphs of Revenue, Cost, and Profit Functions
C
R
Elastic
demand
P
Inelastic
demand
Maximum
profit
Fixed
cost
x
Break-even
point
x
x
Negative of
fixed cost
Revenue Function
Cost Function
The low prices required to
sell more units eventually
result in a decreasing
revenue.
The total cost to produce
x units includes the
fixed cost.
Profit Function
The break-even
point occurs
when R C.
Marginals
dR
marginal revenue
dx
the extra revenue from selling one additional unit
dC
marginal cost
dx
the extra cost of producing one additional unit
dP
marginal profit
dx
the extra profit from selling one additional unit
Marginal
revenue
1 unit
Extra revenue
for one unit
Revenue Function
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APPENDIX B
Formulas from Finance
Basic Terms
P amount of deposit
r interest rate
n number of times interest is compounded per year
t number of years
A balance after t years
Compound Interest Formulas
1. Balance when interest is compounded n times per year
AP 1
r
n
nt
2. Balance when interest is compounded continuously
A Pert
Effective Rate of Interest
reff 1 r
n
n
1
Present Value of a Future Investment
A
P
1 nr nt
Balance of an Increasing Annuity After n Deposits of P per Year for t Years
1 nr AP
nt
1 1
n
r
Initial Deposit for a Decreasing Annuity with n Withdrawals of W per Year for t Years
nr1 1 1rn nt
PW
Monthly Installment M for a Loan of P Dollars over t Years at r% Interest
MP
r12
1
1
1 r12
12t
Amount of an Annuity
T
erT
ctert dt
0
ct is the continuous income function in dollars per year and T is the term of the
annuity in years.
Formulas
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