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 Functions489
- Ch. 8 - Techniques of Integration398
- Ch. 9 - First-Order Differential Equations60
- Ch. 10 - Infinite Sequences and Series119
- 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
Textbook Question
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
Textbook Question
4. Use the properties of logarithms to write the expressions in Exercises 3 and 4 as a single term.
a. ln secθ + ln cosθ
Textbook Question
155. Which is bigger, πᵉ or e^π?
Calculators have taken some of the mystery out of this once-challenging question.
(Go ahead and check; you will see that it is a very close call.)
You can answer the question without a calculator, though.
a. Find an equation for the line through the origin tangent to the graph of
y = ln(x).
Textbook Question
143.
a. Show that ∫ ln(x) dx = x ln(x) − x + C.
Textbook Question
In Exercises 41–44:
a. Find f⁻¹(x).
44. f(x) = 2x², x ≥ 0, a = 5