Flashcards in Rule/Identity Memorization Deck (94):

1

## What is the derivative of u * v? (Product Rule)

### uv' + vu'

2

## What is the derivative of u / v? (Quotient Rule)

### vu' - uv' / v^2

3

## What is the derivative of u^n? (Power Rule)

### nu^(n-1)*u'

4

## What is the derivative of |u|?

### u / |u| * u' where u != 0

5

## What is the derivative of ln u?

### u' / u

6

## What is the derivative of e^u?

### e^u * u'

7

## What is the derivative of log_a_u?

### u' / (ln a) * u

8

## What is the derivative of a^u?

### (ln a) * a^u * u'

9

## What is the derivative of sin u?

### cos u * u'

10

## What is the derivative of cos u?

### -(sin u) * u'

11

## What is the derivative of tan u?

### sec^2 u * u'

12

## What is the derivative of cot u?

### -(csc^2 u) * u'

13

## What is the derivative of sec u?

### (sec u tan u) * u'

14

## What is the derivative of csc u?

### -(csc u cot u) * u'

15

## What is the derivative of arcsin u?

### u' / sqrt(1 - u^2)

16

## What is the derivative of arccos u?

### -u' / sqrt(1 - u^2)

17

## What is the derivative of arctan u?

### u' / 1 + u^2

18

## What is the derivative of arccot u?

### -u' / 1 + u^2

19

## What is the derivative of arcsec u?

### u' / |u| * sqrt(u^2 - 1)

20

## What is the derivative of arccsc u?

### -u' / |u| * sqrt(u^2 -1)

21

## What is the derivative of sinh u?

### (cosh u) * u'

22

## What is the derivative of cosh u?

### (sinh u) * u'

23

## What is the derivative of tanh u?

### (sech^2 u) * u'

24

## What is the derivative of coth u?

### -(csch^2 u) * u'

25

## What is the derivative of sech u?

### -(sech u tanh u) * u'

26

## What is the derivative of csch u?

### -(csch u coth u) * u'

27

## What is the integral of a^u du?

### (1 / ln a) * a^u + C

28

## What is the integral of e^u du?

### e^u + C

29

## What is the integral of sin u du?

### -cos u + C

30

## What is the integral of cos u du?

### sin u + C

31

## What is the integral of tan u du?

### -ln|cos u| + C

32

## What is the integral of cot u du?

### ln |sin u| + C

33

## What is the integral of sec u du?

### ln |sec u + tan u| + C

34

## What is the integral of csc u du?

### -ln |csc u + cot u| + C

35

## What is the integral of sec^2 u du?

### tan u + C

36

## What is the integral of csc^2 u du?

### -cot u + C

37

## What is the integral of sec u tan u du?

### sec u + C

38

## What is the integral of csc u cot u du?

### -csc u + C

39

## What is the integral of du / sqrt(a^2 - u^2)?

### arcsin u/a + C

40

## What is the integral of du / a^2 + u^2?

### 1/a arctan u/a + C

41

## What is the integral of du / u * sqrt(u^2 - a^2)?

### 1/a arcsec |u|/a + C

42

## What is sin 2u?

### 2 sin u cos u

43

## What is cos 2u?

### cos^2 u - sin^2 u = 2 cos^2 u - 1 = 1 - 2 sin^2 u

44

## What is tan 2u?

### 2 tan u / 1 - tan^2 u

45

## What is the sin/cos Pythagorean Identity?

### sin^2 x + cos^2 x = 1

46

## What is the tan/sec Pythagorean Identity?

### 1 + tan^2 x = sec^2 x

47

## What is the cot/csc Pythagorean Identity?

### 1 + cot^2 x = csc^2 x

48

## What are the methods used to find ways to integrate?

### Expanding, Subsitution, Rewriting as the sum of two quotients (x + a / b) = (x/b) + (a/b), Completing the Square, Dividing improper rational function (x^2 / x^2 + 1), Adding and subtracting terms in the numerator (+1 - 1), Trigonometric Identities, Pythagorean Conjugate

49

## What is sinh x?

### e^x - e^-x / 2

50

## What is cosh x?

### e^x + e^-x / 2

51

## What is tanh x?

### sinh x / cosh x

52

## What is the cosh/sinh hyperbolic identity?

### cosh^2 x - sinh^2 x = 1

53

## What is the tanh/sech hyperbolic identity?

### tanh^2 x + sech^2 x = 1

54

## What is the coth/csch hyperbolic identity?

### coth^2 x - csch^2 x = 1

55

## What is sinh 2x?

### 2 sinhx cosh x

56

## What is cosh 2x?

### cosh^2 x + sinh^2 x

57

## What is the derivative of sinh u?

### cosh u * u'

58

## What is the derivative of cosh u?

### sinh u * u'

59

## What is the derivative of tanh u?

### sech^2 u * u'

60

## What is the derivative of coth u?

### -csch^2 u * u'

61

## What is the derivative of sech u?

### -(sech u tanh u) * u'

62

## What is the derivative of csch u?

### -(csch u coth u) * u'

63

## What is the integral of cosh u du?

### sinh u + C

64

## What is the integral of sinh u du?

### cosh u + C

65

## What is the integral of sech^2 u du?

### tanh u + C

66

## What is the integral of csch^2 u du?

### -coth u + C

67

## What is the integral of sech u tanh u du?

### -sech u + C

68

## What is the integral of csch u coth u du?

### -csch u + C

69

## How do you find the particular solution?

### Solve for C, then plug C back into the original equation (non-differential)

70

## What is the initial condition?

### The value of the dependent variable or one of its derivatives provided at initial time t=0. (y = 2 when x = -3)

71

## What is the exponential growth and decay model?

### y = Ce^kt, where C is the initial value of y, k is the proportionality constant. Growth occurs when k > 0 and decay occurs when k < 0.

72

## For integrals of the form x^n e^ax, x^n sin ax, or x^n cos ax which function becomes u and which become dv?

### x^n becomes u, dv = e^ax dx, sin ax dx, or cos ax dx

73

## For integrals of the form x^n ln x dx, x^n arcsin ax dx, or x^n arctan ax dx which function become u and which becomes dv?

### u = ln x, arcsin ax, or arctan ax and dv = x^n dx

74

## For integrals of the form e^ax sin bx or e^ax cos bx, which function becomes u and which becomes dv?

### u = sin bx or cox bx and let dv = e^ax dx

75

## If the power of the sine is odd and positive, what do you do?

### Save one sine factor and convert the remaining factors to cosine. Then expand and integrate.

76

## If the power of the cosine is odd and positive, what do you do?

### Save one cosine factor and convert the remaining factors to sines. Then expand and integrate.

77

## If the powers of both the sine and cosine are even and nonnegative, what do you do?

### Make repeated use of the power reducing identities

78

## When using tabular integration and you return to the original function in the derivative column, what do?

### Instead of going one column down on the right, go straight across and add an integral sign. Integrate this term.

79

## When integrating trig integrals, do you want the degree of sin to be even or odd? Cos even or odd? If both the same, even or odd?

### Sin - odd, cos - odd, both - odd

80

## What are the steps to integrate sin or cos integrals?

### 1. Factor out either sin x or cos x, whichever has the odd power. 2. Using pythagorean theorem, change the remaining factors to the other cofunction. 3. Let u = remaining cofunction (u = sin x or u = cos x) 4. Integrate

81

## If the power of the secant is even and positive, what do?

### Save a secant squared factor and convert the remaining factors to tangent. Then expand and integrate.

82

## If the power of the tangent is odd and positive, what do?

### Save a secant-tangent factor and convert the remaining factors to secants. Then expand and integrate.

83

## If there are no secant factors and the power of tangent is even and positive, what do?

### Convert a tangent-squared factor to a secant-squared factor, then expand and repeat if necessary

84

## If the integral is of the form sec^m dx, where m is odd and positive, what do?

### Use integration by parts, using sec^2 x as dv and sec x as u.

85

## If working with trig integrals with powers of secant and tangent and none of the rules work, what do?

### Convert to sine and cosine.

86

## What is sin mx sin nx?

### 1/2 (cos (m-n)x - cos (m+n)x)

87

## What is sin mx cos nx?

### 1/2 (sin (m-n)x + sin (m+n)x)

88

## What is cos mx cos nx?

### 1/2 (cos (m-n)x + cos (m+n)x)

89

## What is the integral of sqrt (a^2 - u^2) du?

### 1/2 (a^2 arcsin u/a + u*sqrt(a^2 -u^2)) + C

90

## What is the integral of sqrt (u^2 - a^2) du?

### 1/2 (u*sqrt(u^2 - a^2) - a^2*ln |u + sqrt(u^2 - a^2)|) + C where u > a.

91

## What is the integral of sqrt(u^2 + a^2) du?

### 1/2 (u*sqrt(u^2 + a^2) + a^2 * ln |u + sqrt(u^2 + a^2)|) + C

92

## What is the sin^2 u power reducing formula?

### 1 - cos 2u / 2

93

## What is the cos^2 u power reducing formula?

### 1 + cos 2u / 2

94