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AP Calculus AB: 9.3.4 Choosing Effective Function Decompositions

Mathematics6 CardsCreated 3 months ago

This section focuses on how to strategically choose a substitution when performing integration by substitution. It explains that the best choice for 𝑢 is typically a part of the integrand whose derivative is also present, making the integral easier to evaluate. It also covers how to simplify trigonometric expressions using identities and emphasizes that effective decomposition is key to simplifying complex integrals.

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Choosing Effective Function Decompositions

  • Experiment with different choices for u when using integration by substitution. A good choice is one whose derivative is expressed elsewhere in the integrand.

  • When working with integrands that include trigonometric expressions, it is sometimes necessary to rewrite those expressions using trig identities.

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Key Terms

Term
Definition

Choosing Effective Function Decompositions

  • Experiment with different choices for u when using integration by substitution. A good choice is one whose derivative is expressed elsewher...

note

  • When applying integration by substitution to composite
    functions, there may be several choices for u.

  • In the case of a ra...

Which of the following is the best choice for au-substitution for the integral∫2x(x^2−4)^6dx?

u = x ^2 − 4

Which of these expressions is the best choice for making a u-substitution for the integral∫sin^32xcos2xdx?

u = sin 2x

Which of the following expressions creates a working u-substitution that solves the following integral?
∫x^3sinx^2dx

u = x^ 2

What is the best choice for a u-substitution for the integral ∫e^cotx csc^2x dx?

u = cot x

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AP Calculus AB: 9.3.4 Choosing Effective Function Decompositions
TermDefinition

Choosing Effective Function Decompositions

  • Experiment with different choices for u when using integration by substitution. A good choice is one whose derivative is expressed elsewhere in the integrand.

  • When working with integrands that include trigonometric expressions, it is sometimes necessary to rewrite those expressions using trig identities.

note

  • When applying integration by substitution to composite
    functions, there may be several choices for u.

  • In the case of a rational function, the best choice is often the denominator.

  • In this example, du/2 produces the expression in the
    numerator.

  • You may want to express trigonometric integrands in terms of sine and cosine before integrating.

  • Since the denominator has cosx raised to a power, choose u to be cosx. Then –du produces the expression in the numerator.

Which of the following is the best choice for au-substitution for the integral∫2x(x^2−4)^6dx?

u = x ^2 − 4

Which of these expressions is the best choice for making a u-substitution for the integral∫sin^32xcos2xdx?

u = sin 2x

Which of the following expressions creates a working u-substitution that solves the following integral?
∫x^3sinx^2dx

u = x^ 2

What is the best choice for a u-substitution for the integral ∫e^cotx csc^2x dx?

u = cot x