Skip to main content
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 8.8.30
Capitolo 8, Problema 8.8.30

The integrals in Exercises 1–34 converge. Evaluate the integrals without using tables.
∫₂⁴ dt / [t√(t² − 4)]

Guida verificata passo dopo passo
1
Identify the integral to solve: \(\int_{2}^{4} \frac{dt}{t \sqrt{t^{2} - 4}}\).
Recognize that the integrand contains a square root of the form \(\sqrt{t^{2} - a^{2}}\), suggesting a trigonometric substitution. Use the substitution \(t = 2 \sec(\theta)\), where \(a = 2\).
Compute the differential \(dt\) in terms of \(d\theta\): since \(t = 2 \sec(\theta)\), then \(dt = 2 \sec(\theta) \tan(\theta) d\theta\).
Rewrite the integral in terms of \(\theta\) by substituting \(t\), \(dt\), and simplifying the expression under the square root: \(\sqrt{t^{2} - 4} = \sqrt{4 \sec^{2}(\theta) - 4} = 2 \tan(\theta)\).
Change the limits of integration from \(t\) to \(\theta\) using \(t = 2 \sec(\theta)\), then simplify the integral and integrate with respect to \(\theta\).

Risposta video verificata per un problema simile:

Questa soluzione video è stata consigliata dai nostri tutor come utile per risolvere questo problema.
Durata del video:
6m

Concetti chiave

Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.

Improper Integrals and Convergence

Understanding when an integral converges is crucial, especially for integrals with infinite limits or integrands with singularities. In this problem, the integral is definite and the integrand is continuous on [2,4], ensuring convergence. Recognizing convergence allows safe evaluation without concern for divergence.
Video consigliato:
Percorso guidato
11:11
Improper Integrals: Infinite Intervals

Trigonometric Substitution

Trigonometric substitution is a technique used to simplify integrals involving expressions like √(t² − a²). By substituting t = a sec(θ), the radical simplifies using trigonometric identities, making the integral easier to evaluate. This method transforms the integral into a trigonometric integral that can be integrated directly.
Video consigliato:
Percorso guidato
6:04
Introduction to Trigonometric Functions

Integration Techniques for Rational Functions

Integrals involving rational functions combined with radicals often require algebraic manipulation or substitution to simplify. Recognizing the structure of the integrand, such as t in the denominator and a radical in the denominator, guides the choice of substitution and integration steps to find an explicit antiderivative.
Video consigliato:
6:04
Intro to Rational Functions
Pratica correlata
Domanda del libro di testo

In Exercises 69–80, determine whether the improper integral converges or diverges. If it converges, evaluate the integral.

∫₋∞⁰ x² e^(x³) dx

14
views
Domanda del libro di testo

Evaluate the integrals in Exercises 1–14.

∫ (3 dx) / √(1 + 9x²)

27
views
Domanda del libro di testo

Evaluate the integrals in Exercises 1–24 using integration by parts.

∫ t² e^(4t) dt

42
views
Domanda del libro di testo

In Exercises 27–40, use a substitution to change the integral into one you can find in the table. Then evaluate the integral.

∫ dt / (tan(t)√4 - sin^2(t))

21
views
Domanda del libro di testo

In Exercises 35–68, use integration, the Direct Comparison Test, or the Limit Comparison Test to test the integrals for convergence. If more than one method applies, use whatever method you prefer.

∫ from 2 to ∞ of ((x dx) / √(x⁴ - 1))

Domanda del libro di testo

In Exercises 39–48, use an appropriate substitution and then a trigonometric substitution to evaluate the integrals.

∫ (e^{t} dt) / ((1 + e^{2t})^{3/2}) from ln(3/4) to ln(4/3)

37
views