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Back to AI Flashcard MakerMathematics /AP Calculus AB: 9.5.3 Trigonometric Substitution Involving a Definite Integral: Part Two

AP Calculus AB: 9.5.3 Trigonometric Substitution Involving a Definite Integral: Part Two

Mathematics9 CardsCreated 3 months ago

This section builds on the previous part by demonstrating how to complete definite integrals using trigonometric substitution. It emphasizes converting back to the original variable using inverse trig functions before evaluating at the limits and highlights the role of the reference triangle in guiding substitutions.

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Trigonometric Substitution Involving a Definite Integral: Part Two

  • When computing a definite integral using substitution, first ignore the limits of integration and treat the integral like an indefinite integral. Convert back to the original variable before evaluating at the endpoints.

  • The labeled right triangle serves as a dictionary for making trig substitutions.

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Key Terms

Term
Definition

Trigonometric Substitution Involving a Definite Integral: Part Two

  • When computing a definite integral using substitution, first ignore the limits of integration and treat the integral like an indefinite int...

note

  • In the previous lecture you used a trigonometric substitution to evaluate the indefinite integral corresponding to this definite integral. ...


Evaluate ∫1 0 x/√4x−x^2dx

2π/3−√3

Evaluate ∫√3 1 dx/x√x^2+3.

1/√3 ln √2−1 / b2−√3

Evaluate ∫ 4 2 √x^2−4 / x dx

2√3−2π/3

Evaluate ∫0 −1 dx/(5−4x−x^2)^3/2

1/9(2√5/5−√2/4)

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AP Calculus AB: 9.5.3 Trigonometric Substitution Involving a Definite Integral: Part Two
TermDefinition

Trigonometric Substitution Involving a Definite Integral: Part Two

  • When computing a definite integral using substitution, first ignore the limits of integration and treat the integral like an indefinite integral. Convert back to the original variable before evaluating at the endpoints.

  • The labeled right triangle serves as a dictionary for making trig substitutions.

note

  • In the previous lecture you used a trigonometric substitution to evaluate the indefinite integral corresponding to this definite integral. Now it is time to express the answer in terms of x.

  • Notice that to find θ in terms of x you will need to use an inverse trigonometric function. You can use any of the different inverse trig functions, but inverse sine is used here since it is the most frequently seen.

  • Now that you have found the solution of the indefinite integral, you are ready to evaluate the definite integral at the limits of integration.

  • After much algebra, the evaluation of the definite integral is complete.


Evaluate ∫1 0 x/√4x−x^2dx

2π/3−√3

Evaluate ∫√3 1 dx/x√x^2+3.

1/√3 ln √2−1 / b2−√3

Evaluate ∫ 4 2 √x^2−4 / x dx

2√3−2π/3

Evaluate ∫0 −1 dx/(5−4x−x^2)^3/2

1/9(2√5/5−√2/4)

Evaluate ∫ 5 4 1/x^2√25−x^2 dx.

3/100

Evaluate ∫1 2/3 √9x^2−4/x dx.

√5−2sec^−1 3/2

Evaluate ∫1 0 dx/(x^2+2x+2)^2

1/2(tan^−1 2− π/4 −1/10)