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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 7.7.69a
Capitolo 7, Problema 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)

Guida verificata passo dopo passo
1
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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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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Inverse Cosine

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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Integrals Involving Natural Logs: Substitution
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Domanda del libro di testo

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)

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84.a. Find the center of mass of a thin plate of constant density covering the region between the curve y=1/√x and the x-axis from x=1 to x=16.

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4. Which of the following functions grow faster than x² as x→∞? Which grow at the same rate as x²? Which grow slower?

a. x² + √x

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4. Use the properties of logarithms to write the expressions in Exercises 3 and 4 as a single term.

a. ln secθ + ln cosθ

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In Exercises 67–72, you will explore some functions and their inverses together with their derivatives and tangent line approximations at specified points. Perform the following steps using your CAS:

a. Plot the function y=f(x) together with its derivative over the given interval. Explain why you know that f is one-to-one over the interval.

72. y= 2-x-x³, -2 ≤ x ≤ 2, x_0 = 3/2

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154. The linearization of log₃x

a. Find the linearization of

f(x) = log₃xatx = 3.

Then round its coefficients to two decimal places.

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