Textbook Question
In Exercises 41–44:
a. Find f⁻¹(x).
42. f(x) = (x + 2) / (1 − x), a = 1/2
Ch. 7 - Transcendental Functions
Hass15th EditionThomas' CalculusISBN: 9780137616077
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Table of contents
- 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
Chapter 7, Problem 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)
Verified step by step guidance1
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 \( \sinh^{-1} \).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
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.
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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.
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07:01Integrals 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.
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05:43Definition of the Definite Integral
Related Practice
Textbook Question
75. a. Find the open intervals on which the function is increasing and decreasing.
g(x) = x(ln x)²
Textbook Question
145. The linearization of eˣ at x = 0
a. Derive the linear approximation eˣ ≈ 1 + x at x = 0.
Textbook Question
78. Which one is correct, and which one is wrong? Give reasons for your answers.
a. lim (x → 0) (x² - 2x) / (x² - sin x) = lim (x → 0) (2x - 2) / (2x - cos x) = lim (x → 0) 2 / (2 + sin x) = 2 / (2 + 0) = 1
1
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Textbook Question
[Technology Exercise] In Exercises 139–141, find the domain and range of each composite function. Then graph the compositions on separate screens. Do the graphs make sense in each case? Give reasons for your answers. Comment on any differences you see.
139. a. y=arctan(tan x)
Textbook Question
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