MATHEMATICS

1. Appendix G.1 ■ Differentiation and Integration Formulas G1
■ Use differentiation and integration tables to supplement differentiation and integration techniques.
Differentiation Formulas
1. 2. 3.
4. 5. 6.
7. 8. 9.
10. 11. 12.
13. 14. 15.
Integration Formulas
Forms Involving
1. 2.
Forms Involving
3.
4.
5.
6.
7.
8.
9.
10. ͵ 1
u͑a ϩ bu͒
du ϭ
1
a
ln
Խ u
a ϩ buԽϩ C
n 1, 2, 3Ϫ
a2
͑n Ϫ 1͒͑a ϩ bu͒nϪ1΅ ϩ C,͵ u2
͑a ϩ bu͒n
du ϭ
1
b3 ΄
Ϫ1
͑n Ϫ 3͒͑a ϩ bu͒nϪ3
ϩ
2a
͑n Ϫ 2͒͑a ϩ bu͒nϪ2
͵ u2
͑a ϩ bu͒3
du ϭ
1
b3 ΄
2a
a ϩ bu
Ϫ
a2
2͑a ϩ bu͒2
ϩ lnԽa ϩ buԽ΅ ϩ C
͵ u2
͑a ϩ bu͒2
du ϭ
1
b3 ΂bu Ϫ
a2
a ϩ bu
Ϫ 2a lnԽa ϩ buԽ΃ ϩ C
͵ u2
a ϩ bu
du ϭ
1
b3 ΄Ϫ
bu
2
͑2a Ϫ bu͒ ϩ a2
lnԽa ϩ buԽ΅ ϩ C
n 1, 2͵ u
͑a ϩ bu͒n
du ϭ
1
b2 ΄
Ϫ1
͑n Ϫ 2͒͑a ϩ bu͒nϪ2
ϩ
a
͑n Ϫ 1͒͑a ϩ bu͒nϪ1΅ ϩ C,
͵ u
͑a ϩ bu͒2
du ϭ
1
b2 ΂ a
a ϩ bu
ϩ lnԽa ϩ buԽ΃ ϩ C
͵ u
a ϩ bu
du ϭ
1
b2
͑bu Ϫ a lnԽa ϩ buԽ͒ ϩ C
a ؉ bu
͵1
u
du ϭ lnԽuԽ ϩ Cn Ϫ1͵un
du ϭ
unϩ1
n ϩ 1
ϩ C,
un
d
dx
͓csc u͔ ϭ Ϫ͑csc u cot u͒uЈ
d
dx
͓sec u͔ ϭ ͑sec u tan u͒uЈ
d
dx
͓cot u͔ ϭ Ϫ͑csc2
u͒uЈ
d
dx
͓tan u͔ ϭ ͑sec2 u͒uЈ
d
dx
͓cos u͔ ϭ Ϫ͑sin u͒uЈ
d
dx
͓sin u͔ ϭ ͑cos u͒uЈ
d
dx
͓eu͔ ϭ euuЈ
d
dx
͓ln u͔ ϭ
uЈ
u
d
dx
͓x͔ ϭ 1
d
dx
͓un͔ ϭ nunϪ1uЈ
d
dx
͓c͔ ϭ 0
d
dx΄
u
v΅ ϭ
vuЈ Ϫ uvЈ
v2
d
dx
͓uv͔ ϭ uvЈ ϩ vuЈ
d
dx
͓u ± v͔ ϭ uЈ ± vЈ
d
dx
͓cu͔ ϭ cuЈ
G Formulas
G.1 Differentiation and Integration Formulas
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2. Integration Formulas (continued)
11.
12.
13.
Forms Involving
14.
15.
16.
17.
18.
19.
20.
Forms Involving
21.
22.
Forms Involving
23.
24.
25.
26.
27.
28. ͵ 1
uΊu2
ϩ a2
du ϭ
Ϫ1
a
ln
Խa ϩ Ίu2
ϩ a2
u Խϩ C
͵ 1
Ίu2
± a2
du ϭ lnԽu ϩ Ίu2
± a2
Խ ϩ C
͵Ίu2
± a2
u2
du ϭ
ϪΊu2
± a2
u
ϩ lnԽu ϩ Ίu2
± a2
Խ ϩ C
͵Ίu2
ϩ a2
u
du ϭ Ίu2
ϩ a2
Ϫ a ln
Խa ϩ Ίu2
ϩ a2
u Խϩ C
͵u2Ίu2
± a2
du ϭ
1
8
͓u͑2u2
± a2
͒Ίu2
± a2
Ϫ a4
lnԽu ϩ Ίu2
± a2
Խ͔ ϩ C
͵Ίu2
± a2
du ϭ
1
2
͑uΊu2
± a2
± a2
lnԽu ϩ Ίu2
± a2
Խ͒ ϩ C
Ίu2 ± a2, a > 0
n 1͵ 1
͑u2
Ϫ a2
͒n
du ϭ
Ϫ1
2a2
͑n Ϫ 1͒ ΄ u
͑u2
Ϫ a2
͒nϪ1
ϩ ͑2n Ϫ 3͒͵ 1
͑u2
Ϫ a2
͒nϪ1
du΅,
͵ 1
u2
Ϫ a2 du ϭ Ϫ͵ 1
a2
Ϫ u2 du ϭ
1
2a
ln
Խu Ϫ a
u ϩ aԽϩ C
u2
؊ a2
, a > 0
͵ un
Ίa ϩ bu
du ϭ
2
͑2n ϩ 1͒b΂unΊa ϩ bu Ϫ na͵ unϪ1
Ίa ϩ bu
du΃
͵ u
Ίa ϩ bu
du ϭ Ϫ
2͑2a Ϫ bu͒
3b2
Ίa ϩ bu ϩ C
n 1͵Ίa ϩ bu
un
du ϭ
Ϫ1
a͑n Ϫ 1͒΄
͑a ϩ bu͒3͞2
unϪ1
ϩ
͑2n Ϫ 5͒b
2 ͵Ίa ϩ bu
unϪ1
du΅,
͵Ίa ϩ bu
u
du ϭ 2Ίa ϩ bu ϩ a͵ 1
uΊa ϩ bu
du
n 1͵ 1
unΊa ϩ bu
du ϭ
Ϫ1
a͑n Ϫ 1͒ ΄Ίa ϩ bu
unϪ1
ϩ
͑2n Ϫ 3͒b
2 ͵ 1
unϪ1Ίa ϩ bu
du΅,
a > 0͵ 1
uΊa ϩ bu
du ϭ
1
Ίa
ln
ԽΊa ϩ bu Ϫ Ίa
Ίa ϩ bu ϩ ΊaԽϩ C,
͵un Ίa ϩ bu du ϭ
2
b͑2n ϩ 3͒΄un
͑a ϩ bu͒3͞2
Ϫ na͵unϪ1Ίa ϩ bu du΅
Ίa ؉ bu
͵ 1
u2
͑a ϩ bu͒2 du ϭ Ϫ
1
a2 ΄
a ϩ 2bu
u͑a ϩ bu͒
ϩ
2b
a
ln
Խ u
a ϩ buԽ΅ ϩ C
͵ 1
u2
͑a ϩ bu͒
du ϭ Ϫ
1
a΂1
u
ϩ
b
a
ln
Խ u
a ϩ buԽ΃ ϩ C
͵ 1
u͑a ϩ bu͒2 du ϭ
1
a΂ 1
a ϩ bu
ϩ
1
a
ln
Խ u
a ϩ buԽ΃ ϩ C
G2 Appendix G ■ Formulas
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3. 29.
30.
31.
Forms Involving
32.
33.
34. 35.
Forms Involving
36. 37.
38. 39.
40.
Forms Involving
41. 42.
43.
44. 45.
Forms Involving sin or cos
46. 47.
48. 49.
50.
51.
52. 53.
54. ͵un sin u du ϭ Ϫun cos u ϩ n͵unϪ1 cos u du
͵u cos u du ϭ cos u ϩ u sin u ϩ C͵u sin u du ϭ sin u Ϫ u cos u ϩ C
͵cosn
u du ϭ
cosnϪ1
u sin u
n
ϩ
n Ϫ 1
n ͵cosnϪ2
u du
͵sinn u du ϭ Ϫ
sinnϪ1
u cos u
n
ϩ
n Ϫ 1
n ͵sinnϪ2 u du
͵cos2
u du ϭ
1
2
͑u ϩ sin u cos u͒ ϩ C͵sin2
u du ϭ
1
2
͑u Ϫ sin u cos u͒ ϩ C
͵cos u du ϭ sin u ϩ C͵sin u du ϭ Ϫcos u ϩ C
uu
͵͑ln u͒n du ϭ u͑ln u͒n Ϫ n ͵͑ln u͒nϪ1 du͵͑ln u͒2 du ϭ u͓2 Ϫ 2 ln u ϩ ͑ln u͒2͔ ϩ C
n Ϫ1͵un
ln u du ϭ
unϩ1
͑n ϩ 1͒2
͓Ϫ1 ϩ ͑n ϩ 1͒ ln u͔ ϩ C,
͵u ln u du ϭ
u2
4
͑Ϫ1 ϩ 2 ln u͒ ϩ C͵ln u du ϭ u͑Ϫ1 ϩ ln u͒ ϩ C
ln u
͵ 1
1 ϩ enu
du ϭ u Ϫ
1
n
ln͑1 ϩ enu͒ ϩ C
͵ 1
1 ϩ eu
du ϭ u Ϫ ln͑1 ϩ eu͒ ϩ C͵uneu du ϭ uneu Ϫ n͵unϪ1eu du
͵ueu
du ϭ ͑u Ϫ 1͒eu
ϩ C͵eu
du ϭ eu
ϩ C
eu
͵ 1
͑a2
Ϫ u2
͒3͞2 du ϭ
u
a2Ίa2
Ϫ u2
ϩ C͵ 1
u2Ίa2
Ϫ u2
du ϭ
ϪΊa2
Ϫ u2
a2
u
ϩ C
͵ 1
uΊa2
Ϫ u2
du ϭ
Ϫ1
a
ln
Խa ϩ Ίa2
Ϫ u2
u Խϩ C
͵Ίa2
Ϫ u2
u
du ϭ Ίa2
Ϫ u2
Ϫ a ln
Խa ϩ Ίa2
Ϫ u2
u Խϩ C
Ίa2 ؊ u2, a > 0
͵ 1
͑u2
± a2
͒3͞2
du ϭ
±u
a2Ίu2
± a2
ϩ C
͵ 1
u2
Ίu2 ± a2
du ϭ ϯ
Ίu2
± a2
a2
u
ϩ C
͵ u2
Ίu2
± a2
du ϭ
1
2
͑uΊu2
± a2
ϯ a2
lnԽu ϩ Ίu2
± a2
Խ͒ ϩ C
Appendix G.1 ■ Differentiation and Integration Formulas G3
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4. Integration Formulas (continued)
55.
56.
57.
58.
Forms Involving tan , cot , sec , or csc
59.
60.
61.
62.
63.
64.
65.
66.
67.
68.
69.
70.
71.
72.
73.
74. ͵ 1
1 ± csc u
du ϭ u Ϫ tan u ± sec u ϩ C
͵ 1
1 ± sec u
du ϭ u ϩ cot u ϯ csc u ϩ C
͵ 1
1 ± cot u
du ϭ
1
2
͑u ϯ lnԽsin u ± cos uԽ͒ ϩ C
͵ 1
1 ± tan u
du ϭ
1
2
͑u ± lnԽcos u ± sin uԽ͒ ϩ C
n 1͵cscn
u du ϭ Ϫ
cscnϪ2
u cot u
n Ϫ 1
ϩ
n Ϫ 2
n Ϫ 1 ͵cscnϪ2
u du,
n 1͵secn
u du ϭ
secnϪ2
u tan u
n Ϫ 1
ϩ
n Ϫ 2
n Ϫ 1 ͵secnϪ2
u du,
n 1͵cotn
u du ϭ Ϫ
cotnϪ1
u
n Ϫ 1
Ϫ ͵cotnϪ2
u du,
n 1͵tann
u du ϭ
tannϪ1
u
n Ϫ 1
Ϫ ͵tannϪ2
u du,
͵csc2
u du ϭ Ϫcot u ϩ C
͵sec2
u du ϭ tan u ϩ C
͵cot2
u du ϭ Ϫu Ϫ cot u ϩ C
͵tan2
u du ϭ Ϫu ϩ tan u ϩ C
͵csc u du ϭ lnԽcsc u Ϫ cot uԽ ϩ C
͵sec u du ϭ lnԽsec u ϩ tan uԽ ϩ C
͵cot u du ϭ lnԽsin uԽ ϩ C
͵tan u du ϭ ϪlnԽcos uԽ ϩ C
uuuu
͵ 1
sin u cos u
du ϭ lnԽtan uԽ ϩ C
͵ 1
1 ± cos u
du ϭ Ϫcot u ± csc u ϩ C
͵ 1
1 ± sin u
du ϭ tan u ϯ sec u ϩ C
͵un
cos u du ϭ un
sin u Ϫ n͵unϪ1
sin u du
G4 Appendix G ■ Formulas
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5. Appendix G.2 ■ Formulas from Business and Finance G5
■ Summary of business and finance formulas
Formulas from Business
Basic Terms
number of units produced (or sold)
price per unit
total revenue from selling units
total cost of producing units
average cost per unit
total profit from selling units
Basic Equations
Typical Graphs of Supply and Demand Curves
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 Function: price required to sell units
When the demand is inelastic. When the demand is elastic.
Typical Graphs of Revenue, Cost, and Profit Functions
Revenue Function Cost Function Profit Function
The low prices required to The total cost to produce The break-even point occurs
sell more units eventually units includes the fixed when
result in a decreasing cost.
revenue.
R ϭ C.x
Maximum
profit
x
Negative of
fixed cost
Break-even
point
P
x
Fixed
cost
C
Inelastic
demand
x
Elastic
demand
R
͒Խ␩Խ > 1,Խ␩Խ < 1,͑
␩ ϭ
p͞x
dp͞dx
ϭ price elasticity of demand
xp ‫؍‬ fͧxͨ ‫؍‬
x
Demand
Equilibrium point
(x0, p0)
Supply
x0
p0
Equilibrium quantity
Equilibrium
price
p
P ϭ R Ϫ CC ϭ
C
x
R ϭ xp
xP ϭ
C ϭ
xC ϭ
xR ϭ
p ϭ
x ϭ
G.2 Formulas from Business and Finance
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6. Formulas from Business (continued)
Marginals
marginal revenue the extra revenue from selling one additional unit
marginal cost the extra cost of producing one additional unit
marginal profit the extra profit from selling one additional unit
Formulas from Finance
Basic Terms
amount of deposit interest rate
number of times interest is compounded per year
number of years balance after years
Compound Interest Formulas
1. Balance when interest is compounded times per year:
2. Balance when interest is compounded continuously:
Effective Rate of Interest
Present Value of a Future Investment
Balance of an Increasing Annuity After Deposits of per Year for Years
Initial Deposit for a Decreasing Annuity with Withdrawals of per Year for Years
Monthly Installment for a Loan of Dollars over Years at % Interest
Amount of an Annuity
is the continuous income function in dollars per year and is the term of the annuity in years.Tc͑t͒
erT
͵T
0
c͑t͒eϪrt
dt
M ϭ P
Ά
r͞12
1 Ϫ ΄
1
1 ϩ ͑r͞12͒΅
12t
·
rtPM
P ϭ W΂n
r΃Ά1 Ϫ ΄
1
1 ϩ ͑r͞n͒΅
nt
·
tWn
A ϭ P΄΂1 ϩ
r
n΃
nt
Ϫ 1΅΂1 ϩ
n
r΃
tPn
P ϭ
A
΂1 ϩ
r
n΃
nt
reff ϭ ΂1 ϩ
r
n΃
n
Ϫ 1
A ϭ Pert
A ϭ P΂1 ϩ
r
n΃
nt
n
tA ϭt ϭ
n ϭ
r ϭP ϭ
Ϸ
dP
dx
ϭ
Ϸ
dC
dx
ϭ
Ϸ
dR
dx
ϭ
G6 Appendix G ■ Formulas
1 unit
Extra revenue
for one unit
Marginal
revenue
Revenue Function
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