Skip to main content
Ch. 7 - Transcendental Functions
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. 7 - Transcendental FunctionsProblema 7.7.67a
Capitolo 7, Problema 7.7.67a

Evaluate the integrals in Exercises 67–74 in terms of
a. inverse hyperbolic functions.
67. ∫(from 0 to 2√3)dx/√(4+x²)

Guida verificata passo dopo passo
1
Recognize that the integral \( \int \frac{dx}{\sqrt{a^2 + x^2}} \) can be expressed in terms of the inverse hyperbolic sine function, \( \sinh^{-1}(x/a) \), where \( a \) is a constant.
Identify the constant \( a \) in the integral. Here, the integral is \( \int_0^{2\sqrt{3}} \frac{dx}{\sqrt{4 + x^2}} \), so \( a^2 = 4 \) which means \( a = 2 \).
Rewrite the integral using the formula: \( \int \frac{dx}{\sqrt{a^2 + x^2}} = \sinh^{-1}\left( \frac{x}{a} \right) + C \). For the definite integral, evaluate \( \sinh^{-1}\left( \frac{x}{2} \right) \) from 0 to \( 2\sqrt{3} \).
Set up the evaluation: compute \( \sinh^{-1}\left( \frac{2\sqrt{3}}{2} \right) - \sinh^{-1}\left( \frac{0}{2} \right) \). Simplify the arguments inside the inverse hyperbolic sine functions.
Recall that \( \sinh^{-1}(y) = \ln\left(y + \sqrt{y^2 + 1}\right) \) if you want to express the answer in logarithmic form, but since the problem asks for inverse hyperbolic functions, leave the answer in terms of \( \sinh^{-1} \).

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:
2m

Concetti chiave

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

Inverse Hyperbolic Functions

Inverse hyperbolic functions, such as sinh⁻¹(x) and cosh⁻¹(x), are the inverses of hyperbolic sine and cosine functions. They often appear in integrals involving expressions like √(a² + x²). Recognizing these forms allows rewriting integrals in terms of inverse hyperbolic functions for simpler evaluation.
Video consigliato:
4:49
Inverse Cosine

Integration of Rational Functions Involving Square Roots

Integrals containing expressions like 1/√(a² + x²) are common and can be solved using substitution or by recalling standard integral formulas. These integrals typically result in inverse hyperbolic functions, making it essential to identify the pattern to apply the correct formula efficiently.
Video consigliato:
07:01
Integrals Involving Natural Logs: Substitution

Definite Integration and Evaluation of Limits

Definite integrals require evaluating the antiderivative at the upper and lower limits. After finding the integral in terms of inverse hyperbolic functions, substituting the limits correctly and simplifying the result is crucial to obtain the final numerical value.
Video consigliato:
Percorso guidato
05:43
Definition of the Definite Integral
Pratica correlata
Domanda del libro di testo

In Exercises 41–44:

a. Find f⁻¹(x).


41. f(x) = 2x + 3, a = −1

26
views
Domanda del libro di testo

Suppose that the function g and its derivative with respect to x have the following values at x=0, 1, 2, 3, and 4.

Assuming the inverse function g^(-1) is differentiable, find the slope of g^(-1)(x) at

a. x=1

34
views
Domanda del libro di testo

Use reference triangles in an appropriate quadrant to find the angles in Exercises 1–8.

7. a. sec^(-1)(-√2)

26
views
Domanda del libro di testo

145. The linearization of eˣ at x = 0

a. Derive the linear approximation eˣ ≈ 1 + x at x = 0.

33
views
Domanda del libro di testo

21. a. Show that ln(x) grows slower as x→∞ than x^(1/n) for any positive integer n, even x^(1/1,000,000).

28
views
Domanda del libro di testo

77. Which one is correct, and which one is wrong? Give reasons for your answers.

a. lim (x → 3) (x - 3) / (x² - 3) = lim (x → 3) 1 / (2x) = 1/6

20
views