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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 8.8.18
Capitolo 8, Problema 8.8.18

The integrals in Exercises 1–34 converge. Evaluate the integrals without using tables.
∫₁^∞ dx / [x√(x² − 1)]

Guida verificata passo dopo passo
1
Identify the integral to be evaluated: \(\displaystyle \int_1^{\infty} \frac{dx}{x \sqrt{x^2 - 1}}\).
Recognize that the integrand involves \(\sqrt{x^2 - 1}\), which suggests a trigonometric substitution such as \(x = \sec(\theta)\), because \(\sec^2(\theta) - 1 = \tan^2(\theta)\).
Perform the substitution \(x = \sec(\theta)\), then compute \(dx = \sec(\theta) \tan(\theta) d\theta\). Also, rewrite the integrand in terms of \(\theta\):
\[\frac{1}{x \sqrt{x^2 - 1}} dx = \frac{1}{\sec(\theta) \sqrt{\sec^2(\theta) - 1}} \cdot \sec(\theta) \tan(\theta) d\theta.\]
Simplify the expression inside the integral using the identity \(\sqrt{\sec^2(\theta) - 1} = \tan(\theta)\), and then simplify the integrand to a function of \(\theta\) that is easier to integrate.
Change the limits of integration from \(x\) to \(\theta\) using \(x = \sec(\theta)\), then integrate with respect to \(\theta\). Finally, substitute back to \(x\) to express the answer in terms of the original variable.

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Improper Integrals

Improper integrals involve integration over an infinite interval or integrands with infinite discontinuities. To evaluate them, we replace the infinite limit with a variable, compute the integral, and then take the limit as the variable approaches infinity to determine convergence and value.
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Improper Integrals: Infinite Intervals

Substitution Method

The substitution method simplifies integrals by changing variables to transform the integrand into a more manageable form. Choosing an appropriate substitution, such as a trigonometric or hyperbolic function, can help evaluate integrals involving expressions like √(x² − 1).
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Euler's Method

Trigonometric Identities and Inverse Functions

Trigonometric identities relate expressions involving squares and roots, such as x² − 1, to trigonometric functions like secant and tangent. Recognizing these allows rewriting the integral in terms of inverse trigonometric functions, facilitating evaluation without tables.
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Derivatives of Other Inverse Trigonometric Functions
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