Domanda del libro di testo
In Exercises 13–24, find the derivative of y with respect to the appropriate variable.
17. y = ln(sinh z)
25
views
Ch. 7 - Transcendental Functions
Hass15th EditionThomas' CalculusISBN: 9780137616077
Non è quello che usi tu?Cambia libro di testoScegli un libro di testo diverso
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 7, Problema 7.6.65
Evaluate the integrals in Exercises 53–76.
65. ∫3dr/√(1-4(r-1)²)
Guida verificata passo dopo passo1
Recognize that the integral has the form \( \int \frac{3 \, dr}{\sqrt{1 - 4(r-1)^2}} \), which resembles the standard integral \( \int \frac{dx}{\sqrt{a^2 - x^2}} = \arcsin\left(\frac{x}{a}\right) + C \).
Make a substitution to simplify the expression inside the square root. Let \( u = r - 1 \), so that \( du = dr \). The integral becomes \( \int \frac{3 \, du}{\sqrt{1 - 4u^2}} \).
Rewrite the expression under the square root as \( 1 - (2u)^2 \), which suggests another substitution or recognizing \( a = 1 \) and \( x = 2u \).
To match the standard form, factor out the constant inside the square root: \( \sqrt{1 - 4u^2} = \sqrt{1 - (2u)^2} \). Consider substituting \( w = 2u \), so \( dw = 2 du \) or \( du = \frac{dw}{2} \).
Rewrite the integral in terms of \( w \) and use the standard arcsine integral formula: \( \int \frac{dw}{\sqrt{1 - w^2}} = \arcsin(w) + C \). After integrating, substitute back \( w = 2u \) and \( u = r - 1 \) to express the answer in terms of \( r \).
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:
3mConcetti chiave
Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.
Integration of Functions Involving Square Roots
This concept involves integrating functions that contain square roots, often requiring substitution or recognizing standard integral forms. Understanding how to manipulate the integrand to match known integral formulas is essential for solving these problems.
Video consigliato:
07:01
Integrals Involving Natural Logs: Substitution
Trigonometric Substitution
Trigonometric substitution is a technique used to simplify integrals involving expressions like √(a² - x²), √(x² - a²), or √(a² + x²). By substituting x with a trigonometric function, the integral becomes easier to evaluate using trigonometric identities.
Video consigliato:
Percorso guidato
6:04Introduction to Trigonometric Functions
Definite and Indefinite Integrals
Understanding the difference between definite and indefinite integrals is crucial. Indefinite integrals represent families of functions plus a constant, while definite integrals compute the net area under a curve between limits. This problem involves an indefinite integral, focusing on finding the antiderivative.
Video consigliato:
Percorso guidato
05:43Definition of the Definite Integral
Pratica correlata
Domanda del libro di testo
Evaluate the integrals in Exercises 97–110.
99. ∫₀³ (√2 + 1)x^(√2) dx
26
views
Domanda del libro di testo
Find the values in Exercises 9–12.
9. sin(arccos((√2)/2))
26
views
Domanda del libro di testo
In Exercises 115–126, use logarithmic differentiation or the method in Example 6 to find the derivative of y with respect to the given independent variable.
115. y = (x + 1)ˣ
37
views
Domanda del libro di testo
Each of Exercises 1–4 gives a value of sinh x or cosh x. Use the definitions and the identity cosh²x - sinh²x = 1 to find the values of the remaining five hyperbolic functions.
2. sinh x = 4/3
35
views
Domanda del libro di testo
In Exercises 13–24, find the derivative of y with respect to the appropriate variable.
21. y = ln(cosh v) - 1/2 tanh²v
24
views