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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 8.4.12
Capitolo 8, Problema 8.4.12

Evaluate the integrals in Exercises 1–14.
∫ √(y² - 25) / y³ dy, where y > 5

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Identify the integral to solve: \(\int \frac{\sqrt{y^{2} - 25}}{y^{3}} \, dy\), with the condition \(y > 5\).
Recognize that the integrand contains a square root of the form \(\sqrt{y^{2} - a^{2}}\), which suggests using a trigonometric substitution. Since \(y > 5\), set \(y = 5 \sec(\theta)\), where \(\theta\) is in the appropriate domain to keep \(y > 5\).
Compute the differential \(dy\) in terms of \(d\theta\): \(dy = 5 \sec(\theta) \tan(\theta) \, d\theta\).
Rewrite the integral in terms of \(\theta\) by substituting \(y = 5 \sec(\theta)\) and \(dy\) as above. Simplify the expression inside the square root and the powers of \(y\) accordingly.
Simplify the resulting integral using trigonometric identities, then integrate with respect to \(\theta\). After integration, substitute back \(\theta = \sec^{-1}(\frac{y}{5})\) to express the answer in terms of \(y\).

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Integration of Rational Functions Involving Radicals

This concept involves integrating functions where the integrand contains radicals combined with rational expressions. Techniques often include algebraic manipulation or substitution to simplify the integrand into a more manageable form for integration.
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Trigonometric Substitution

Trigonometric substitution is a method used to evaluate integrals containing expressions like √(y² - a²). By substituting y = a sec(θ), the radical simplifies using trigonometric identities, making the integral easier to solve.
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Definite Domain Considerations and Variable Restrictions

When integrating functions with radicals, the domain restrictions (e.g., y > 5) ensure the expression under the square root is non-negative. Recognizing these constraints is essential to choose the correct substitution and interpret the integral's result properly.
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Definition of the Definite Integral
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