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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 8.6.14
Capitolo 8, Problema 8.6.14

Use the table of integrals at the back of the text to evaluate the integrals in Exercises 1–26.
∫ √(x² - 4) / x dx

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1
Recognize that the integral is of the form \(\int \frac{\sqrt{x^2 - a^2}}{x} \, dx\) where \(a = 2\). This suggests using a trigonometric substitution to simplify the square root expression.
Use the substitution \(x = 2 \sec \theta\), which implies \(dx = 2 \sec \theta \tan \theta \, d\theta\). This substitution transforms \(\sqrt{x^2 - 4}\) into \(\sqrt{4 \sec^2 \theta - 4} = 2 \tan \theta\).
Rewrite the integral in terms of \(\theta\) by substituting \(x\), \(dx\), and \(\sqrt{x^2 - 4}\) with their expressions involving \(\theta\). The integral becomes \(\int \frac{2 \tan \theta}{2 \sec \theta} \cdot 2 \sec \theta \tan \theta \, d\theta\).
Simplify the integrand by canceling terms and combining like factors. This will reduce the integral to a function involving trigonometric expressions in \(\theta\) that are easier to integrate.
Integrate the simplified trigonometric expression with respect to \(\theta\), then substitute back \(\theta = \sec^{-1}(x/2)\) to express the answer in terms of \(x\).

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Integration Using Standard Integral Formulas

Many integrals can be evaluated by recognizing their form and applying standard integral formulas from tables. These tables list integrals of common functions, including those involving roots and rational expressions, which simplifies the integration process without performing substitution or integration by parts manually.
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Integration by Parts for Definite Integrals

Trigonometric Substitution

Trigonometric substitution is a technique used to simplify integrals involving expressions like √(x² - a²). By substituting x = a sec(θ), the radical simplifies to a trigonometric function, making the integral easier to evaluate. This method is often referenced in integral tables for such forms.
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Introduction to Trigonometric Functions

Algebraic Manipulation of the Integrand

Before applying integral formulas, it is important to manipulate the integrand algebraically to match a standard form. This may involve factoring, simplifying fractions, or rewriting expressions to fit the integral forms listed in the table, ensuring the correct formula is applied.
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Completing the Square to Rewrite the Integrand
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