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 8.4.8
Capitolo 8, Problema 8.4.8

Evaluate the integrals in Exercises 1–14.
∫ √(1 - 9t²) dt

Guida verificata passo dopo passo
1
Recognize that the integral \( \int \sqrt{1 - 9t^2} \, dt \) resembles the form \( \int \sqrt{a^2 - u^2} \, du \), which suggests using a trigonometric substitution to simplify the square root expression.
Set the substitution \( u = 3t \), so that \( u^2 = 9t^2 \). Then, rewrite the integral in terms of \( u \): \( \int \sqrt{1 - u^2} \cdot \frac{du}{3} \). This simplifies the integral to \( \frac{1}{3} \int \sqrt{1 - u^2} \, du \).
Use the trigonometric substitution \( u = \sin \theta \), which implies \( du = \cos \theta \, d\theta \). This transforms the integral into \( \frac{1}{3} \int \sqrt{1 - \sin^2 \theta} \cdot \cos \theta \, d\theta \).
Simplify the square root using the Pythagorean identity: \( \sqrt{1 - \sin^2 \theta} = \cos \theta \). The integral becomes \( \frac{1}{3} \int \cos \theta \cdot \cos \theta \, d\theta = \frac{1}{3} \int \cos^2 \theta \, d\theta \).
Use the power-reduction formula for \( \cos^2 \theta \): \( \cos^2 \theta = \frac{1 + \cos 2\theta}{2} \). Substitute this into the integral and integrate with respect to \( \theta \). After integrating, substitute back \( \theta = \arcsin u \) and then \( u = 3t \) to express the answer in terms of \( t \).

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:
9m

Concetti chiave

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

Trigonometric Substitution

Trigonometric substitution is a technique used to evaluate integrals involving square roots of quadratic expressions. By substituting a trigonometric function for the variable, the integral simplifies using trigonometric identities. For example, for integrals with √(a² - x²), substituting x = a sin(θ) helps transform the integral into a trigonometric form.
Video consigliato:
Percorso guidato
6:04
Introduction to Trigonometric Functions

Integration of Trigonometric Functions

After substitution, the integral often involves trigonometric functions like sine and cosine. Understanding how to integrate these functions, including using identities and standard integral formulas, is essential. This allows the integral to be evaluated in terms of θ before converting back to the original variable.
Video consigliato:
Percorso guidato
6:04
Introduction to Trigonometric Functions

Back-Substitution

Once the integral is evaluated in terms of the trigonometric variable, back-substitution replaces the trigonometric expression with the original variable. This step uses the initial substitution relationship and trigonometric identities to express the final answer in terms of the original variable t.
Video consigliato:
04:27
Substitution With an Extra Variable
Pratica correlata
Domanda del libro di testo

Evaluate the integrals in Exercises 33–52.

∫ sec(x) tan²(x) dx

28
views
Domanda del libro di testo

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

∫ dy / (y√(1 + (ln y)²)) from 1 to e

39
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 -∞ to ∞ of ((dx) / (e^x + e^(-x)))

46
views
Domanda del libro di testo

The integrals in Exercises 1–44 are in no particular order. Evaluate each integral using any algebraic method, trigonometric identity, or substitution you think is appropriate.

∫ (6 dy / √y(1 + y))

27
views
Domanda del libro di testo

Use any method to evaluate the integrals in Exercises 55–66.

∫ x² √(1 - x²) dx

21
views
Domanda del libro di testo

Evaluate the integrals in Exercises 33–52.

∫ sec³(x) tan³(x) dx

33
views