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Ch. 8 - Techniques of Integration
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoHass 15th EditionCh. 8 - Techniques of IntegrationProblema 8.PE.31b
Capitolo 8, Problema 8.PE.31b

Evaluate the integrals in Exercises 29–32 (b) using a trigonometric substitution.
∫ [x / √(4 − x²)] dx

Guida verificata passo dopo passo
1
Identify the form of the integral: the integrand contains \( \sqrt{4 - x^2} \), which suggests using the trigonometric substitution \( x = 2 \sin(\theta) \) because \( 4 - x^2 = 4 - 4\sin^2(\theta) = 4\cos^2(\theta) \).
Compute the differential \( dx \) in terms of \( d\theta \): since \( x = 2 \sin(\theta) \), then \( dx = 2 \cos(\theta) d\theta \).
Rewrite the integral in terms of \( \theta \): substitute \( x = 2 \sin(\theta) \), \( dx = 2 \cos(\theta) d\theta \), and \( \sqrt{4 - x^2} = 2 \cos(\theta) \) into the integral \( \int \frac{x}{\sqrt{4 - x^2}} dx \).
Simplify the integral expression after substitution: the integral becomes \( \int \frac{2 \sin(\theta)}{2 \cos(\theta)} \times 2 \cos(\theta) d\theta \), which simplifies to \( \int 2 \sin(\theta) d\theta \).
Integrate with respect to \( \theta \): find \( \int 2 \sin(\theta) d\theta \), then substitute back \( \theta = \arcsin(\frac{x}{2}) \) to express the answer in terms of \( x \).

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Trigonometric Substitution

Trigonometric substitution is a technique used to simplify integrals involving square roots of quadratic expressions by substituting a trigonometric function for the variable. For expressions like √(a² - x²), substituting x = a sin(θ) transforms the integral into a trigonometric form that is easier to evaluate.
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Introduction to Trigonometric Functions

Integration of Trigonometric Functions

After substitution, the integral often involves trigonometric functions such as sine and cosine. Understanding how to integrate these functions, including using identities and basic integral formulas, is essential to solve the integral and then revert back to the original variable.
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Introduction to Trigonometric Functions

Back Substitution

Once the integral is evaluated in terms of the trigonometric variable, back substitution is used to rewrite the answer in terms of the original variable x. This involves using the inverse trigonometric functions or right triangle relationships derived from the substitution.
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Substitution With an Extra Variable
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∫ [1 / (x(9 - x²))] dx

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Evaluate the integrals in Exercises 33–36.

∫ [1 / √(9 - x²)] dx

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