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Calculus 2 > Rule/Identity Memorization > Flashcards

Rule/Identity Memorization Flashcards

1
Q

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

A

uv’ + vu’

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2
Q

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

A

vu’ - uv’ / v^2

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3
Q

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

A

nu^(n-1)*u’

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4
Q

What is the derivative of |u|?

A

u / |u| * u’ where u != 0

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5
Q

What is the derivative of ln u?

A

u’ / u

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6
Q

What is the derivative of e^u?

A

e^u * u’

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7
Q

What is the derivative of log_a_u?

A

u’ / (ln a) * u

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8
Q

What is the derivative of a^u?

A

(ln a) * a^u * u’

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9
Q

What is the derivative of sin u?

A

cos u * u’

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10
Q

What is the derivative of cos u?

A

-(sin u) * u’

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11
Q

What is the derivative of tan u?

A

sec^2 u * u’

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12
Q

What is the derivative of cot u?

A

-(csc^2 u) * u’

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13
Q

What is the derivative of sec u?

A

(sec u tan u) * u’

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14
Q

What is the derivative of csc u?

A

-(csc u cot u) * u’

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15
Q

What is the derivative of arcsin u?

A

u’ / sqrt(1 - u^2)

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16
Q

What is the derivative of arccos u?

A

-u’ / sqrt(1 - u^2)

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17
Q

What is the derivative of arctan u?

A

u’ / 1 + u^2

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18
Q

What is the derivative of arccot u?

A

-u’ / 1 + u^2

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19
Q

What is the derivative of arcsec u?

A

u’ / |u| * sqrt(u^2 - 1)

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20
Q

What is the derivative of arccsc u?

A

-u’ / |u| * sqrt(u^2 -1)

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21
Q

What is the derivative of sinh u?

A

(cosh u) * u’

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22
Q

What is the derivative of cosh u?

A

(sinh u) * u’

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23
Q

What is the derivative of tanh u?

A

(sech^2 u) * u’

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24
Q

What is the derivative of coth u?

A

-(csch^2 u) * u’

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25
Q

What is the derivative of sech u?

A

-(sech u tanh u) * u’

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26
Q

What is the derivative of csch u?

A

-(csch u coth u) * u’

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27
Q

What is the integral of a^u du?

A

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

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28
Q

What is the integral of e^u du?

A

e^u + C

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29
Q

What is the integral of sin u du?

A

-cos u + C

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30
Q

What is the integral of cos u du?

A

sin u + C

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31
Q

What is the integral of tan u du?

A

-ln|cos u| + C

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32
Q

What is the integral of cot u du?

A

ln |sin u| + C

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33
Q

What is the integral of sec u du?

A

ln |sec u + tan u| + C

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34
Q

What is the integral of csc u du?

A

-ln |csc u + cot u| + C

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35
Q

What is the integral of sec^2 u du?

A

tan u + C

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36
Q

What is the integral of csc^2 u du?

A

-cot u + C

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37
Q

What is the integral of sec u tan u du?

A

sec u + C

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38
Q

What is the integral of csc u cot u du?

A

-csc u + C

39
Q

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

A

arcsin u/a + C

40
Q

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

A

1/a arctan u/a + C

41
Q

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

A

1/a arcsec |u|/a + C

42
Q

What is sin 2u?

A

2 sin u cos u

43
Q

What is cos 2u?

A

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

44
Q

What is tan 2u?

A

2 tan u / 1 - tan^2 u

45
Q

What is the sin/cos Pythagorean Identity?

A

sin^2 x + cos^2 x = 1

46
Q

What is the tan/sec Pythagorean Identity?

A

1 + tan^2 x = sec^2 x

47
Q

What is the cot/csc Pythagorean Identity?

A

1 + cot^2 x = csc^2 x

48
Q

What are the methods used to find ways to integrate?

A

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
Q

What is sinh x?

A

e^x - e^-x / 2

50
Q

What is cosh x?

A

e^x + e^-x / 2

51
Q

What is tanh x?

A

sinh x / cosh x

52
Q

What is the cosh/sinh hyperbolic identity?

A

cosh^2 x - sinh^2 x = 1

53
Q

What is the tanh/sech hyperbolic identity?

A

tanh^2 x + sech^2 x = 1

54
Q

What is the coth/csch hyperbolic identity?

A

coth^2 x - csch^2 x = 1

55
Q

What is sinh 2x?

A

2 sinhx cosh x

56
Q

What is cosh 2x?

A

cosh^2 x + sinh^2 x

57
Q

What is the derivative of sinh u?

A

cosh u * u’

58
Q

What is the derivative of cosh u?

A

sinh u * u’

59
Q

What is the derivative of tanh u?

A

sech^2 u * u’

60
Q

What is the derivative of coth u?

A

-csch^2 u * u’

61
Q

What is the derivative of sech u?

A

-(sech u tanh u) * u’

62
Q

What is the derivative of csch u?

A

-(csch u coth u) * u’

63
Q

What is the integral of cosh u du?

A

sinh u + C

64
Q

What is the integral of sinh u du?

A

cosh u + C

65
Q

What is the integral of sech^2 u du?

A

tanh u + C

66
Q

What is the integral of csch^2 u du?

A

-coth u + C

67
Q

What is the integral of sech u tanh u du?

A

-sech u + C

68
Q

What is the integral of csch u coth u du?

A

-csch u + C

69
Q

How do you find the particular solution?

A

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

70
Q

What is the initial condition?

A

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

71
Q

What is the exponential growth and decay model?

A

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
Q

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?

A

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

73
Q

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?

A

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

74
Q

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

A

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

75
Q

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

A

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

76
Q

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

A

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

77
Q

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

A

Make repeated use of the power reducing identities

78
Q

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

A

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

79
Q

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?

A

Sin - odd, cos - odd, both - odd

80
Q

What are the steps to integrate sin or cos integrals?

A
  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
Q

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

A

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

82
Q

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

A

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

83
Q

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

A

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

84
Q

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

A

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

85
Q

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

A

Convert to sine and cosine.

86
Q

What is sin mx sin nx?

A

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

87
Q

What is sin mx cos nx?

A

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

88
Q

What is cos mx cos nx?

A

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

89
Q

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

A

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

90
Q

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

A

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

91
Q

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

A

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

92
Q

What is the sin^2 u power reducing formula?

A

1 - cos 2u / 2

93
Q

What is the cos^2 u power reducing formula?

A

1 + cos 2u / 2

94
Q

What is the tan^2 u power reducing formula?

A

1 - cos 2u / 1 + cos 2u

Calculus 2 (2 decks)

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