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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 8.1.54f
Capitolo 8, Problema 8.1.54f

Using different substitutions
Show that the integral
∫((x² - 1)(x + 1))^(-2/3) dx
can be evaluated with any of the following substitutions.
f. u = arccos x
What is the value of the integral?

Guida verificata passo dopo passo
1
Start with the integral \( \int ((x^{2} - 1)(x + 1))^{-\frac{2}{3}} \, dx \). First, simplify the expression inside the integral. Notice that \( x^{2} - 1 = (x - 1)(x + 1) \), so the integrand becomes \( ((x - 1)(x + 1)(x + 1))^{-\frac{2}{3}} = ((x - 1)(x + 1)^{2})^{-\frac{2}{3}} \).
Rewrite the integrand as \( (x - 1)^{-\frac{2}{3}} (x + 1)^{-\frac{4}{3}} \) by distributing the exponent \( -\frac{2}{3} \) to each factor.
Use the substitution \( u = \arccos x \). Then, \( x = \cos u \) and \( dx = -\sin u \, du \). This substitution is useful because it relates \( x \) to trigonometric functions, which can simplify expressions involving \( x^{2} - 1 \).
Express the factors \( x - 1 \) and \( x + 1 \) in terms of \( u \): \( x - 1 = \cos u - 1 \) and \( x + 1 = \cos u + 1 \). Also, note that \( \sin^{2} u = 1 - \cos^{2} u \), which can help simplify the expression further.
Rewrite the integral entirely in terms of \( u \) and \( du \), substituting \( x \), \( dx \), and the factors \( x - 1 \), \( x + 1 \). Then simplify the resulting integral, which should be easier to evaluate using standard trigonometric integral techniques.

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Trigonometric Substitution

Trigonometric substitution is a technique used to simplify integrals involving algebraic expressions by substituting a trigonometric function for a variable. For example, substituting u = arccos(x) transforms expressions involving x into trigonometric forms, often making the integral easier to evaluate by leveraging trigonometric identities.
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Introduction to Trigonometric Functions

Integration of Powers of Functions

Integrating expressions raised to fractional powers, such as ((x² - 1)(x + 1))^(-2/3), requires understanding how to manipulate and simplify the integrand. This often involves rewriting the expression in a more manageable form or using substitution to convert it into a standard integral form.
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Representing Functions as Power Series

Inverse Trigonometric Functions and Their Derivatives

Inverse trigonometric functions like arccos(x) have specific derivatives that are essential when performing substitution in integrals. Knowing that d/dx[arccos(x)] = -1/√(1 - x²) helps in changing variables and adjusting the differential dx accordingly during integration.
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Derivatives of Other Inverse Trigonometric Functions
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Using different substitutions

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∫((x² - 1)(x + 1))^(-2/3) dx

can be evaluated with any of the following substitutions.

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What is the value of the integral?

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The instructions for the integrals in Exercises 1–10 have three parts, one for the Midpoint Rule, one for the Trapezoidal Rule, and one for Simpson’s Rule.

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b. Evaluate the integral directly and find |ET|.

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