Domanda del libro di testo
The integrals in Exercises 1–34 converge. Evaluate the integrals without using tables.
∫₋∞^∞ (x dx) / (x² + 4)^(3/2)
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Ch. 8 - Techniques of Integration
Hass15th EditionThomas' CalculusISBN: 9780137616077
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TOC.tableOfContents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions490
- Ch. 8 - Techniques of Integration399
- Ch. 9 - First-Order Differential Equations66
- Ch. 10 - Infinite Sequences and Series132
- Ch. 11 - Parametric Equations and Polar Coordinates58
Capitolo 8, Problema 8.4.14
Evaluate the integrals in Exercises 1–14.
∫ (2 dx) / (x³ √(x² - 1)), where x > 1
Guida verificata passo dopo passo1
Identify the integral to solve: \(\int \frac{2 \, dx}{x^{3} \sqrt{x^{2} - 1}}\) with the condition \(x > 1\).
Recognize that the integrand contains \(\sqrt{x^{2} - 1}\), which suggests using a trigonometric substitution such as \(x = \sec(\theta)\) because \(\sec^{2}(\theta) - 1 = \tan^{2}(\theta)\).
Perform the substitution: let \(x = \sec(\theta)\), then compute \(dx = \sec(\theta) \tan(\theta) \, d\theta\). Also, rewrite the expressions inside the integral in terms of \(\theta\).
Rewrite the integral entirely in terms of \(\theta\) by substituting \(x\), \(dx\), and \(\sqrt{x^{2} - 1}\), then simplify the resulting expression to a form that is easier to integrate.
Integrate with respect to \(\theta\), then back-substitute using \(\theta = \sec^{-1}(x)\) to express the answer in terms of \(x\).
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Integration Techniques for Rational Functions
This involves methods to integrate functions expressed as ratios of polynomials or involving roots. Recognizing the form helps decide whether substitution, partial fractions, or trigonometric substitution is appropriate to simplify the integral.
Video consigliato:
6:04
Intro to Rational Functions
Trigonometric Substitution
Trigonometric substitution is used to simplify integrals containing expressions like √(x² - a²). By substituting x = a sec(θ), the radical simplifies using trigonometric identities, making the integral easier to evaluate.
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Percorso guidato
6:04Introduction to Trigonometric Functions
Domain Considerations and Restrictions
Understanding the domain (x > 1) is crucial because it affects the choice of substitution and the sign of expressions like √(x² - 1). It ensures the substitution is valid and the integral is evaluated correctly within the given constraints.
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Percorso guidato
5:10Finding the Domain and Range of a Graph
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Evaluate the integrals in Exercises 39–54.
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Domanda del libro di testo
The integrals in Exercises 1–34 converge. Evaluate the integrals without using tables.
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Domanda del libro di testo
Evaluate the integrals in Exercises 1–22.
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