AP Calculus BC > Unit 06 Part 2: Integration Rules > Flashcards

Unit 06 Part 2: Integration Rules Flashcards

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1
Q

Natural Exponential Function (e^x)

A

The inverse of the natural logarithmic function (ln x). The base “e” is a mathematical constant, not a variable.

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2
Q

Domain of e^x

A

The set of all real numbers (-infinity to infinity).

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3
Q

Range of e^x

A

The set of all positive real numbers (0 to infinity).

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4
Q

One-to-One Function

A

A property of the natural exponential function meaning that for every output, there is exactly one input.

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5
Q

Inverse Properties of e and ln

A

ln(e^x) = x and e^(ln x) = x. They effectively “cancel” each other out.

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6
Q

Relative Extrema

A

The local maximum and minimum values of a function.

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7
Q

Critical Numbers

A

The x-values where the derivative of a function is equal to zero or undefined; used to find relative extrema.

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8
Q

Constant of Integration (C)

A

A constant added to the end of an indefinite integral to represent the family of functions that could have the same derivative.

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9
Q

Base “a”

A

Refers to an exponential or logarithmic base that is any positive real number other than e (e.g., 2, 4, 10, etc.).

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10
Q

Derivative of a^x

A

The rate of change of an exponential function with base a, defined as ln(a) * a^x.

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11
Q

Derivative of log_a(x)

A

The rate of change of a logarithmic function with base a, defined as 1 / (ln(a) * x).

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12
Q

Integral of a^x

A

The antiderivative of an exponential function with base a, defined as (a^x / ln(a)) + C.

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13
Q

Change of Base Formula

A

A logarithmic property used to convert a general logarithm into a natural logarithm: log_a(x) = ln(x) / ln(a).

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14
Q

Composite Function

A

A function containing another function inside it (e.g., 4^(7x)), requiring the Chain Rule for differentiation.

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15
Q

Exact Solution

A

A specific function result where the Constant of Integration (C) is solved for using a given initial value point.

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16
Q

Antidifferentiation Rules (Inverse Trig)

A

The process of going backwards from a derivative to original functions that result in inverse trig functions.

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17
Q

Sine Inverse (arcsin) Integral Rule

A

The function resulting from the integral of 1 / sqrt(1 - x^2) dx.

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18
Q

Tangent Inverse (arctan) Integral Rule

A

The function resulting from the integral of 1 / (1 + x^2) dx.

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19
Q

Secant Inverse (arcsec) Integral Rule

A

The function resulting from the integral of 1 / (x * sqrt(x^2 - 1)) dx. Result involves the absolute value of x.

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20
Q

Pattern Recognition in Integration

A

The primary strategy used to identify which inverse trigonometric rule applies to a specific problem.

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21
Q

Algebraic Manipulation

A

Rearranging or breaking apart an expression (like a denominator) to fit a known integration pattern.

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22
Q

Power-Reducing Identity: sin^2 x

A

sin^2 x = (1 - cos 2x) / 2

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23
Q

Power-Reducing Identity: cos^2 x

A

cos^2 x = (1 + cos 2x) / 2

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24
Q

Pythagorean Identity: sin^2 x

A

sin^2 x = 1 - cos^2 x

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25
Pythagorean Identity: cos^2 x
cos^2 x = 1 - sin^2 x
26
U-Substitution (Trig)
The primary integration technique used after factoring out a single trig term and converting the remaining terms.
27
Case 1: Odd Sine
Strategy: Factor out one sin x for du; convert remaining sin terms to cosine; use u = cos x.
28
Case 2: Odd Cosine
Strategy: Factor out one cos x for du; convert remaining cos terms to sine; use u = sin x.
29
Case 3: Even Powers (Sine/Cosine)
Strategy: Use power-reducing identities to replace squared terms and then integrate individually.
30
Trigonometric Substitution
A technique to evaluate integrals containing radicals by substituting the variable with a trigonometric function.
31
Radical Expression Forms (Trig Sub)
Algebraic expressions containing roots, specifically sqrt(a^2 - x^2) or sqrt(x^2 + a^2).
32
Trig Identity: 1 - sin^2(theta)
cos^2(theta)
33
Trig Identity: tan^2(theta) + 1
sec^2(theta)
34
Right Triangle Method
A geometric approach to convert the answer from "theta" back to "x" by labeling triangle sides (Opp/Adj/Hyp).
35
Constant (a) in Trig Sub
A fixed numerical value (often as a^2) that determines the coefficients of the trigonometric substitution.
36
Integration by Parts
A technique to integrate the product of two functions; the "reverse" of the Product Rule.
37
The Product Rule
d/dx [u * v] = u(dv/dx) + v(du/dx); the basis for Integration by Parts.
38
u (Integration by Parts)
The part of the integrand chosen to be differentiated.
39
dv (Integration by Parts)
The part of the integrand (including dx) chosen to be integrated.
40
Tabular Method
A shorthand organizational strategy for repeated integration by parts using columns for signs, u, and dv.
41
Partial Fractions
Technique to decompose a complex rational function into a sum of simpler fractions.
42
Rational Function
A function that is the ratio of two polynomials (numerator over denominator).
43
Linear Factor
A factor in the denominator where the variable is to the first power (e.g., x - 2).
44
Repeated Factor
A factor in the denominator raised to a power higher than one (e.g., (x - 2)^2).
45
Improper Integral
A definite integral with an infinite limit or a discontinuous integrand on the interval.
46
Divergence
Property where the area under an improper integral grows without bound (limit does not exist/equals infinity).
47
Convergence
Property where the limit of an improper integral exists and results in a finite numerical value.
48
Discontinuous Integrand
A function not continuous at a point within the limits, such as a vertical asymptote.
49
Solving for an Exponent
Take the natural log (ln) of both sides of the equation.
50
Solving for a Natural Log
Exponentiate both sides (make both sides exponents with base e).
51
Derivative of e^u (Chain Rule)
d/dx [e^u] = e^u * du/dx
52
Steps to find Relative Extrema of e^x
1. Find derivative; 2. Set to zero to find critical numbers; 3. Perform number line test.
53
Integral of e^u du
e^u + C
54
Derivative of a^u (Chain Rule)
ln(a) * a^u * u'
55
Derivative of log_a(u) (Chain Rule)
1 / (ln(a) * u) * u'
56
Integrating a^x Rule
(a^x / ln(a)) + C
57
Integrating Logarithms (Change of Base)
Rewrite log_a(x) as ln(x)/ln(a); move 1/ln(a) outside; use u-sub (u=ln x).
58
Integral of 1 / sqrt(1 - x^2) dx
arcsin(x) + C
59
Integral of 1 / (1 + x^2) dx
arctan(x) + C
60
Integral of 1 / (x * sqrt(x^2 - 1)) dx
arcsec|x| + C
61
Completing the Square in Integration
An algebraic step used to manipulate a denominator to fit an inverse trig pattern (e.g., arctan).
62
Sine Substitution (Trig Sub)
Use x = a * sin(theta) for the form sqrt(a^2 - x^2).
63
Tangent Substitution (Trig Sub)
Use x = a * tan(theta) for the form sqrt(x^2 + a^2).
64
Secant Substitution (Trig Sub)
Use x = a * sec(theta) for the form sqrt(x^2 - a^2).
65
Integration by Parts Formula
Integral of u dv = uv - Integral of v du
66
LIATE/Selection Criteria for u
Choose u such that it becomes simpler when differentiated; choose dv such that it is easy to integrate.
67
Tabular Method Setup
Three columns: 1. Alternating Signs (+/-); 2. u and its derivatives; 3. dv and its integrals.
68
Partial Fraction Setup: Distinct Linear Factors
A / (factor 1) + B / (factor 2)
69
Partial Fraction Setup: Repeated Factors
Create a fraction for every power up to the exponent (e.g., A/(x-2) + B/(x-2)^2).
70
Solving for Partial Fraction Constants
Pick values of x that make terms zero (Strategic Substitution) or pick easy values like x=0.
71
Evaluating Infinite Limits
Replace infinity with a variable (b), evaluate the integral, and take the limit as b approaches infinity.
72
Splitting Improper Integrals
For limits from -inf to +inf, split the integral at a middle point (usually 0) and solve two separate limits.
73
Limit of arctan(x) as x approaches infinity
pi / 2
74
Limit of arctan(x) as x approaches -infinity
-pi / 2
75
General Integration Strategy (Trig)
Always look for an odd power first; it is generally more straightforward than even-power cases.
AP Calculus BC
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