Calculus: Techniques of Integration flashcards

Question 1

What formula does integration by parts use?

Accepted Answer

∫u dv = uv − ∫v du

Question 2

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

Accepted Answer

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

Question 3

Does integration by parts apply to definite integrals, indefinite integrals, or both?

Accepted Answer

Both

Question 4

What is a trigonometric integral?

Accepted Answer

An integral of products and powers of trig functions

Question 5

Product-to-sum identity: sin(ax)sin(bx) = ?

Accepted Answer

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

Question 6

Product-to-sum identity: sin(ax)cos(bx) = ?

Accepted Answer

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

Question 7

Product-to-sum identity: cos(ax)cos(bx) = ?

Accepted Answer

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

Question 8

What does a power reduction formula do?

Accepted Answer

It swaps the integral of a trig function raised to a power for one with a lower power

Question 9

What does trigonometric substitution turn an algebraic integral with √(a² − x²), √(a² + x²) or √(x² − a²) into?

Accepted Answer

A trigonometric integral

Question 10

An integrand contains √(a² − x²): which substitution is used?

Accepted Answer

x = a sin θ, with dx = a cos θ dθ

Question 11

Which substitution suits an integral involving √(a² + x²)?

Accepted Answer

x = a tan θ, with dx = a sec²θ dθ

Question 12

Which trig substitution handles √(x² − a²)?

Accepted Answer

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

Question 13

What is partial fraction decomposition?

Accepted Answer

A technique that splits a rational function into a sum of simpler rational functions

Question 14

Before partial fraction decomposition, what must be true of the degrees, and what do you do if it is not?

Accepted Answer

The numerator's degree must be less than the denominator's; otherwise do long division first

Question 15

What is an integration table?

Accepted Answer

A table listing integration formulas

Question 16

What is a computer algebra system (CAS)?

Accepted Answer

Technology that carries out many mathematical tasks, integration among them

Question 17

What is numerical integration?

Accepted Answer

The range of numerical methods for estimating a definite integral, such as the midpoint, trapezoidal and Simpson's rules

Question 18

Midpoint rule: M_n = ?

Accepted Answer

The sum from i = 1 to n of f(m_i)Δx, where m_i is the midpoint of the ith subinterval

Question 19

What shapes does the trapezoidal rule use to approximate a definite integral?

Accepted Answer

Trapezoids

Question 20

Trapezoidal rule: T_n = ?

Accepted Answer

(1/2)Δx[f(x₀) + 2f(x₁) + 2f(x₂) + ⋯ + 2f(xₙ₋₁) + f(xₙ)]

Question 21

What kind of function does Simpson's rule integrate piece by piece to approximate a definite integral?

Accepted Answer

A piecewise quadratic function

Question 22

Simpson's rule: S_n = ?

Accepted Answer

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

Question 23

Error bound for the midpoint rule M_n?

Accepted Answer

Error ≤ M(b − a)³/(24n²)

Question 24

Error bound for the trapezoidal rule T_n?

Accepted Answer

Error ≤ M(b − a)³/(12n²)

Question 25

Error bound for Simpson's rule S_n?

Accepted Answer

Error ≤ M(b − a)⁵/(180n⁴)

Question 26

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

Accepted Answer

|A − B|

Question 27

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

Accepted Answer

|(A − B)/A|, often given as a percentage: |(A − B)/A| · 100%

Question 28

What makes an integral improper?

Accepted Answer

An infinite interval of integration, or an infinite discontinuity of the integrand on the interval

Question 29

When does an improper integral converge?

Accepted Answer

When its defining limit is a finite real number; otherwise it diverges

Question 30

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

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