AP Calculus: Techniques of Integration flashcards

Question 1

What is the integration by parts formula?

Accepted Answer

∫ u dv = uv − ∫ v du

Question 2

What is integration by parts for a definite integral from a to b?

Accepted Answer

∫ from a to b of u dv = [uv] from a to b − ∫ from a to b of v du

Question 3

If B estimates a quantity whose actual value is A, what is the absolute error?

Accepted Answer

|A − B|

Question 4

If B estimates a quantity whose actual value is A, what is the relative error as a percentage?

Accepted Answer

|(A − B)/A| · 100%

Question 5

What is an improper integral?

Accepted Answer

An integral over an infinite interval, or of a function with an infinite discontinuity on the interval; it is defined as a limit.

Question 6

When does an improper integral converge?

Accepted Answer

When the limit that defines it is a finite real number; otherwise it diverges.

Question 7

How is ∫ from a to +∞ of f(x) dx defined?

Accepted Answer

As the limit, as t → +∞, of ∫ from a to t of f(x) dx

Question 8

How is ∫ from −∞ to b of f(x) dx defined?

Accepted Answer

As the limit, as t → −∞, of ∫ from t to b of f(x) dx

Question 9

What does partial fraction decomposition do?

Accepted Answer

It breaks a rational function down into a sum of simple rational functions.

Question 10

Before partial fraction decomposition, what must be true of the numerator's degree?

Accepted Answer

It must be less than the denominator's degree; if not, do long division first.

Question 11

Which substitution is used for integrals involving √(a² − x²)?

Accepted Answer

x = a sin θ, with dx = a cos θ dθ

Question 12

Which substitution is used for integrals involving √(a² + x²)?

Accepted Answer

x = a tan θ, with dx = a sec² θ dθ

Question 13

Which substitution is used for integrals involving √(x² − a²)?

Accepted Answer

x = a sec θ, with dx = a sec θ tan θ dθ

Question 14

What is sin(ax) sin(bx) rewritten as a sum?

Accepted Answer

(1/2) cos((a − b)x) − (1/2) cos((a + b)x)

Question 15

What is sin(ax) cos(bx) rewritten as a sum?

Accepted Answer

(1/2) sin((a − b)x) + (1/2) sin((a + b)x)

Question 16

What is cos(ax) cos(bx) rewritten as a sum?

Accepted Answer

(1/2) cos((a − b)x) + (1/2) cos((a + b)x)

Question 17

What is the reduction formula for ∫ secⁿx dx (n ≠ 1)?

Accepted Answer

(secⁿ⁻²x tan x)/(n − 1) + ((n − 2)/(n − 1)) ∫ secⁿ⁻²x dx

Question 18

What is the midpoint rule approximation Mₙ?

Accepted Answer

Mₙ = Σ from i = 1 to n of f(mᵢ) Δx, where mᵢ is the midpoint of the ith subinterval

Question 19

What is the trapezoidal rule approximation Tₙ?

Accepted Answer

Tₙ = (1/2) Δx (f(x₀) + 2f(x₁) + 2f(x₂) + ⋯ + 2f(xₙ₋₁) + f(xₙ))

Question 20

What is Simpson's rule approximation Sₙ?

Accepted Answer

Sₙ = (Δx/3)(f(x₀) + 4f(x₁) + 2f(x₂) + 4f(x₃) + ⋯ + 2f(xₙ₋₂) + 4f(xₙ₋₁) + f(xₙ))

Question 21

Simpson's rule approximates a function with what kind of pieces?

Accepted Answer

Piecewise quadratic functions

Question 22

What is a computer algebra system (CAS)?

Accepted Answer

Technology that performs many mathematical tasks, including integration

Question 23

What is numerical integration?

Accepted Answer

Numerical methods for estimating a definite integral's value, such as the midpoint, trapezoidal and Simpson's rules

AP Calculus: Techniques of Integration flashcards

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