Differentiation Formulas

■ 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. 冕 ^u

^{a ⫹ 1 bu}

^{du ⫽ 1 a ln} ⱍ ^{a ⫹ u bu} ⱍ ^{⫹ C}

n ⫽ 1, 2, 3

⫺ a

共n

⫺ 1兲共a ⫹ bu兲

^{⫹ C,}

冕

^{⫹ u}

^bu兲

^{du ⫽ b 1}

^{⫺ 3兲共a ⫺ 1 ⫹ bu兲}

^⫹

^{⫺ 2兲共a 2a ⫹ bu兲}

冕

^{a ⫹ u}

^bu

^{du ⫽ b 1}

^{a ⫹ 2a bu ⫺ 2}

^{a ⫹ a}

^bu

^{⫹ ln ⱍ a ⫹ bu ⱍ}

^{⫹ C} 冕

^{⫹ u}

^bu兲

^{du ⫽ b 1}

^{bu ⫺ a ⫹ a}

^{bu ⫺ 2a ln ⱍ a ⫹ bu ⱍ}

^{⫹ C} 冕 ^{a ⫹ u}

^{bu du ⫽ b 1}

^{⫺ bu 2}

^{2a ⫺ bu}

^{⫹ a}

^{ln ⱍ a ⫹ bu ⱍ}

^{⫹ C}

n ⫽ 1, 2

冕

^{⫹ u bu兲}

^{du ⫽ b 1}

^{⫺ 2兲共a ⫺ 1 ⫹ bu兲}

^⫹

^{⫺ 1兲共a a ⫹ bu兲}

^{⫹ C,} 冕

^{a ⫹ u bu}

^{du ⫽ b 1}

^{a ⫹ a bu ⫹ ln ⱍ a ⫹ bu ⱍ}

^{⫹ C}

冕 ^{a ⫹ u bu du ⫽}

^{b 1}

^{⫺ a ln ⱍ a ⫹ bu ⱍ}

^{⫹ C}

冕 ^{1 u du ⫽ ln ⱍ u ⱍ ⫹ C}

n ⫽ ⫺ 1

冕 ^u

^{du ⫽ n u}

^⫹

^{1 ⫹ C,}

n

d

dx

csc u

⫽ ⫺共 csc u cot u

u ⬘ d

dx

sec u

⫽

sec u tan u

u ⬘ d

dx

cot u

⫽ ⫺共 csc

u

u ⬘

d

dx

⫽

u兲u ⬘ d

dx

⫽ ⫺

⬘ d

dx

sin u兴 ⫽

⬘

d

dx

e

⫽ e

u ⬘ d

dx

ln u

⫽ u ⬘ u d

dx

x

⫽ 1

d

dx

⫽ nu

u ⬘ d

dx

⫽ 0 d

dx

^{u v}

^{⫽ vu ⬘ ⫺ v}

^{uv ⬘}

d

dx

uv

⫽ uv ⬘ ⫹ vu ⬘ d

dx

u

v

⫽ u ⬘

v ⬘ d

dx

cu

⫽ cu ⬘

G ^Formulas

G.1 Differentiation and Integration Formulas

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. 冕 ^{冕 u}

^u

^{1 ⫹ a}

^{du ⫽ ⫺ a 1 ln} ⱍ ^{a ⫹}

^{u u}

^{⫹ a}

ⱍ ^{⫹ C}

1

冪

u

a

du ⫽ ln ⱍ ^{u ⫹}

^u

^a

ⱍ ^{⫹ C}

冕

^u

^u

^a

^{du ⫽ ⫺}

^{u u}

^a

^{⫹ ln ⱍ u ⫹}

^u

^a

^{ⱍ ⫹ C} 冕

^u

^{u ⫹ a}

^{du ⫽}

^u

^{⫹ a}

^{⫺ a ln} ⱍ ^{a ⫹}

^{u u}

^{⫹ a}

ⱍ ^{⫹ C}

冕 ^u

^u

^a

^{du ⫽ 1 8}

^u

^2u

^a

^u

^a

^{⫺ a}

^{ln ⱍ u ⫹}

^u

^a

^ⱍ

^{⫹ C} 冕

^u

^a

^du

^⫽

^{1 2}

^u

^u

^a

^a

^{ln ⱍ u ⫹}

^u

^a

^ⱍ

^{⫹ C}

2±a², a >

0

n ⫽ 1

冕

^{⫺ 1 a}

^{du ⫽ 2a}

^{⫺ 1 ⫺ 1兲}

^{⫺ u a}

^⫹

^{2n ⫺ 3}

冕

^{⫺ 1 a}

^du

^, 冕 ^u

^{⫺ 1 a}

^{du ⫽ ⫺}

冕 ^a

^{⫺ 1 u}

^{du ⫽ 2a 1 ln} ⱍ ^{u u ⫺ ⫹ a a} ⱍ ^{⫹ C}

2 ⴚ a², a >

0

冕

^{a u} ⫹

bu du ⫽ 2

共2n

⫹ 1兲b

^u

^{a ⫹ bu ⫺ na} 冕

^{u a}

⫹

bu du

冕

^{a u} ⫹ bu du ⫽ ⫺ 2共2a ⫺ bu兲

3b

a ⫹ bu ⫹ C

n ⫽ 1

冕

^{a u ⫹}

^{bu du ⫽ a共n ⫺ ⫺ 1 1兲}

^{a ⫹ u}

^bu

^⫹

^{2n ⫺ 2 5}

^b 冕

^{a u}

^{⫹ bu du}

^, 冕

^{a u ⫹ bu du ⫽ 2}

^{a ⫹ bu ⫹ a} 冕 ^u

^{a 1} ⫹ bu du

n ⫽ 1

冕 ^u

^{a 1} ⫹ bu du ⫽ ⫺ 1

a共n ⫺ 1兲

^{a u}

^{⫹ bu ⫹}

^{2n ⫺ 2 3}

^b 冕 ^u

^{1 a} ⫹ bu du

^,

a

0

冕 ^u

^{a 1} ⫹ bu du ⫽ 1

冪

a ln ⱍ

^{a a ⫹ ⫹ bu bu ⫺ ⫹}

^{a a} ⱍ ^{⫹ C,}

冕 ^u

^{a ⫹ bu du}

^⫽

^{b共2n 2 ⫹ 3兲}

^u

^{a ⫹ bu}

^{⫺ na} 冕 ^u

^{a ⫹ bu du}

冕 ^u

^{⫹ 1 bu兲}

^{du ⫽ ⫺ a 1}

^{u共a a ⫹ ⫹ 2bu bu兲 ⫹ 2b a ln} ⱍ ^{a ⫹ u bu} ⱍ

^{⫹ C}

冕 ^u

^{a 1 ⫹ bu}

^{du ⫽ ⫺ 1 a}

^{1 u ⫹ b a ln} ⱍ ^{a ⫹ u bu} ⱍ

^{⫹ C}

冕 ^{u共a ⫹ 1 bu兲}

^{du ⫽ 1 a}

^{a ⫹ 1 bu ⫹ 1 a ln} ⱍ ^{a ⫹ u bu} ⱍ

^{⫹ C}

G2 Appendix G

Formulas

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. 冕 冕 ^u u sin u du

^{sin u du} ⫽ ^{⫽ ⫺} sin u ^u

^{cos u} ⫺ u cos u ^{⫹ n} 冕 ⫹ ^u C

^{cos u du} 冕 u cos u du ⫽ cos u ⫹ u sin u ⫹ C

冕 ^cos

^{u du ⫽ cos}

^{n u sin u ⫹ n ⫺ n 1} 冕 ^cos

^{u du}

冕 ^sin

^{u du ⫽ ⫺ sin}

^{n u cos u ⫹ n ⫺ n 1} 冕 ^sin

^{u du} 冕 ^cos

^{u du ⫽ 1 2}

^{u ⫹} sin u cos u

⫹ C

冕 ^sin

^{u du ⫽ 1 2}

^{u ⫺} sin u cos u

⫹ C

冕 ^{cos u du ⫽ sin u ⫹ C} 冕 ^{sin u du ⫽ ⫺ cos u}

^{⫹ C}

冕 ^{共ln u兲}

^{du ⫽ u共ln u兲}

^{⫺ n} 冕 ^{共ln u兲}

^du 冕 ^{共ln u兲}

^{du ⫽ u关2 ⫺ 2 ln u ⫹}

^{⫹ C}

n ⫽ ⫺ 1

冕 ^u

^{ln u du ⫽}

^{n u ⫹}

¹

^{1 ⫹}

^{n ⫹ 1}

^{ln u}

^{⫹ C,} 冕 ^{u ln u du ⫽ u 4}

^{共⫺ 1 ⫹ 2 ln u兲 ⫹ C} 冕 ^{ln u du ⫽ u共⫺ ln u 1 ⫹ ln u兲 ⫹ C}

冕 ^{1 ⫹ 1 e}

^{du ⫽ u ⫺ 1 n ln共1 ⫹ e}

^{⫹ C} 冕 ^{1 ⫹ 1 e}

^{du ⫽ u ⫺ ln}

^{1 ⫹ e}

^{⫹ C} 冕 ^u

^e

^{du ⫽ u}

^e

^{⫺ n} 冕 ^u

^e

^du 冕 ^ue

^{du ⫽}

^{⫺ 1兲e}

^{⫹ C}

冕 ^e

^{du ⫽ e}

^⫹

^C

u

冕

^{⫺ 1 u}

^{du ⫽ a}

^{a u}

⫺ u

⫹ C

冕 ^u

^{a 1}

⫺ u

du ⫽ ⫺

a

⫺ u

a

u ⫹ C

冕 ^u

^a

¹ ⫺ u

du ⫽ ⫺ 1

a ln ⱍ ^{a ⫹}

^{u a}

^{⫺ u}

ⱍ ^{⫹ C}

冕

^a

^{u ⫺ u}

^du

^⫽

^a

^{⫺ u}

^{⫺ a ln} ⱍ ^{a ⫹}

^{u a}

^{⫺ u}

ⱍ ^{⫹ C}

2ⴚu², a >

0

冕

^{1 a}

^{du ⫽ a}

^u

^u

a

⫹ C

冕 ^u

^{u 1}

^a

^{du ⫽ ⫿}

^{u a}

^{u a}

^{⫹ C} 冕

^u

a

⫽

2

⫿ ⱍ ^⫹ ⱍ

^⫹

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. 冕 冕 冕 冕 ^{1 1 1 1}

^{1 1 1 1 csc u sec u cot u tan u du du du du ⫽ ⫽ ⫽ ⫽ 1 2 1 2 u u}

^{⫺ ⫹ u ⫿}

^{tan u cot u ln ln ⱍ ⱍ sin u cos u}

^{⫿ csc u sec u}

^{cos u sin u ⫹ ⫹ C C ⱍ ⱍ}

^{⫹ ⫹ C C}

n ⫽ 1

冕 ^csc

^{u du ⫽ ⫺ csc}

ⁿ

^{⫺ u cot u 1 ⫹ n n ⫺ ⫺ 2 1} 冕 ^csc

^{u du,}

n ⫽ 1

冕 ^sec

^{u du ⫽ sec}

^{n ⫺ u tan u 1 ⫹ n n ⫺ ⫺ 2 1} 冕 ^sec

^{u du,}

n ⫽ 1

冕 ^cot

^{u du ⫽ ⫺ cot n}

^⫺

^{1 u ⫺} 冕 ^cot

^{u du,}

n ⫽ 1

冕 ^tan

^{u du ⫽ tan n}

^⫺

^{1 u ⫺} 冕 ^tan

^{u du,} 冕 ^csc

^{u du ⫽ ⫺ cot u ⫹ C}

冕 ^sec

^{u du ⫽ tan u ⫹ C} 冕 ^cot

^{u du ⫽ ⫺ u ⫺ cot u ⫹ C} 冕 ^tan

^{u du ⫽ ⫺ u ⫹ tan u ⫹ C} 冕 ^{csc u du ⫽ ln ⱍ csc u ⫺ cot u ⱍ ⫹ C} 冕 ^{sec u du ⫽ ln ⱍ sec u ⫹ tan u ⱍ ⫹ C} 冕 ^{cot u du ⫽ ln ⱍ sin u ⱍ ⫹ C}

冕 ^{tan u du ⫽ ⫺ ln ⱍ}

^{cos u ⱍ ⫹}

^C

冕 sin u cos u ¹ du ⫽ ln ⱍ ^{tan u} ⱍ ^{⫹ C}

冕 ¹

^{1 cos u du ⫽ ⫺ cot u}

^{csc u ⫹ C} 冕 ¹

^{1 sin u du ⫽ tan u ⫿ sec u ⫹ C} 冕 ^u

^{cos u du ⫽ u}

^{sin u ⫺ n} 冕 ^u

^{sin u du}

G4 Appendix G

Formulas

■ 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⫽C.

x

Maximum profit

x Negative of

fixed cost Break-even point P

x Fixed

cost C Inelastic

demand

x Elastic

demand R

ⱍ ^␩ ⱍ

^1,

ⱍ ^␩ ⱍ

^1,

共

␩ ⫽ p

x

dp

dx ⫽ price elasticity of demand

x pⴝf冇x冈ⴝ

x Demand

Equilibrium point (x₀, p₀) Supply

x₀ p₀

Equilibrium quantity Equilibrium

price p

P ⫽ R ⫺ C C ⫽ C

R ⫽ xp x

x P ⫽

C ⫽

x C ⫽

x R ⫽

p ⫽ x ⫽

G.2 Formulas from Business and Finance

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. T c共t兲

e

冕

^c共t兲e

^dt

M ⫽ P

冦 ^{1 ⫺}

^{1 r ⫹}

^{12 1}

^r

¹²

冧

r t

P M

P ⫽ W

^{n r}

^{1 ⫺}

^{1 ⫹ 1}

^r

ⁿ

t W

n

A ⫽ P

^{1 ⫹} n ^r

^{⫺ 1}

^{1 ⫹ n} r

t P

n

P ⫽ A

冢

^{1 ⫹ n r}

r

⫽

^{1 ⫹ r n}

^{⫺ 1}

A ⫽ Pe

A ⫽ P

^{1 ⫹ n r}

n

t A ⫽

t ⫽ n ⫽

r ⫽ P ⫽

dP

dx ⫽ dC

dx ⫽ dR

dx ⫽

G6 Appendix G

Formulas

1 unit

Extra revenue for one unit

Marginal revenue

Revenue Function