Textbook Question
80. Find all values of c that satisfy the conclusion of Cauchy's Mean Value Theorem for the given functions and interval.
a. f(x) = x, g(x) = x², (a, b) = (-2, 0)
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 7.7.69a
Evaluate the integrals in Exercises 67–74 in terms of
a. inverse hyperbolic functions.
69. ∫(from 5/4 to 2)dx/(1-x²)
Verified step by step guidance1
Recognize that the integral \( \int \frac{dx}{1 - x^2} \) can be rewritten using partial fractions because the denominator factors as \( (1 - x)(1 + x) \).
Express the integrand as partial fractions: \( \frac{1}{1 - x^2} = \frac{A}{1 - x} + \frac{B}{1 + x} \). Solve for constants \( A \) and \( B \).
Recall that the integral of \( \frac{1}{1 - x^2} \) can also be expressed in terms of inverse hyperbolic functions, specifically \( \tanh^{-1}(x) \), since \( \frac{d}{dx} \tanh^{-1}(x) = \frac{1}{1 - x^2} \) for \( |x| < 1 \).
Set up the definite integral from \( x = \frac{5}{4} \) to \( x = 2 \) and write the antiderivative in terms of \( \tanh^{-1}(x) \) or equivalently \( \frac{1}{2} \ln \left| \frac{1 + x}{1 - x} \right| \).
Evaluate the antiderivative at the upper and lower limits and subtract to find the value of the definite integral in terms of inverse hyperbolic functions.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Definite Integrals
A definite integral calculates the net area under a curve between two specific limits. It is represented as ∫ from a to b of f(x) dx, where a and b are the lower and upper bounds. Evaluating definite integrals involves finding the antiderivative and then applying the Fundamental Theorem of Calculus.
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Definition of the Definite Integral
Inverse Hyperbolic Functions
Inverse hyperbolic functions, such as arsinh, arcosh, and artanh, are the inverses of hyperbolic functions. They often appear as antiderivatives of rational functions involving expressions like 1 - x². Recognizing when to express integrals in terms of these functions simplifies evaluation and interpretation.
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Integration of Rational Functions Involving 1 - x²
Integrals of the form ∫ dx/(1 - x²) can be decomposed using partial fractions or recognized as derivatives of inverse hyperbolic functions. Specifically, ∫ dx/(1 - x²) relates to the inverse hyperbolic tangent function, artanh(x), which helps in expressing the integral in closed form.
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Related Practice
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Textbook Question
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Verify the integration formulas in Exercises 37–40.
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