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Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBriggs 3rd EditionCh. 7 - Logarithmic, Exponential Functions, and Hyperbolic FunctionsProblema 7.3.38
Capitolo 7, Problema 7.3.38

37–56. Integrals Evaluate each integral.


∫ sech² w tanh w dw

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Recall the definitions and derivatives of hyperbolic functions: \( \frac{d}{dw} \tanh w = \text{sech}^2 w \) and \( \frac{d}{dw} \text{sech} w = -\text{sech} w \tanh w \).
Notice that the integrand is \( \text{sech}^2 w \tanh w \), which can be seen as \( \text{sech}^2 w \times \tanh w \). Consider if this expression resembles the derivative of a product or a function involving \( \tanh w \) or \( \text{sech} w \).
Try to express the integrand in terms of a derivative of a function. For example, check if it matches the derivative of \( -\text{sech} w \) because \( \frac{d}{dw}(-\text{sech} w) = \text{sech} w \tanh w \), which is close but not exactly the integrand.
Alternatively, use substitution: let \( u = \tanh w \), then \( du = \text{sech}^2 w \, dw \). Rewrite the integral in terms of \( u \) and \( du \).
After substitution, the integral becomes \( \int u \, du \), which is straightforward to integrate. Then, substitute back \( u = \tanh w \) to express the answer in terms of \( w \).

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Hyperbolic Functions

Hyperbolic functions, such as sech (hyperbolic secant) and tanh (hyperbolic tangent), are analogs of trigonometric functions but based on hyperbolas. Understanding their definitions and properties is essential for manipulating and integrating expressions involving these functions.
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Asymptotes of Hyperbolas

Integration Techniques for Hyperbolic Functions

Integrating products of hyperbolic functions often involves recognizing derivatives of one function within the integrand or using substitution. Familiarity with derivatives like d/dx(tanh x) = sech² x helps simplify and evaluate such integrals efficiently.
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Substitution Method

The substitution method involves changing variables to simplify an integral. By identifying a part of the integrand as a derivative of another function, substitution transforms the integral into a basic form that is easier to evaluate.
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Euler's Method
Pratica correlata
Domanda del libro di testo

63–68. Definite integrals Evaluate the following definite integrals. Use Theorem 7.7 to express your answer in terms of logarithms.

∫₁/₈¹ dx/x√(1 + x²/³)

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37–56. Integrals Evaluate each integral.

∫ dx/x√(16 + x²)

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29–62. Integrals Evaluate the following integrals. Include absolute values only when needed.


∫ (x²) / (4x³ + 7) dx

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88–91. Limits Use l’Hôpital’s Rule to evaluate the following limits.


lim x → ∞ (1 − coth x) / (1 − tanh x)

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Probability as an integral Two points P and Q are chosen randomly, one on each of two adjacent sides of a unit square (see figure). What is the probability that the area of the triangle formed by the sides of the square and the line segment PQ is less than one-fourth the area of the square? Begin by showing that x and y must satisfy xy < 1/2 in order for the area condition to be met. Then argue that the required probability is: 1/2 + ∫[1/2 to 1] (dx / 2x) and evaluate the integral.

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Domanda del libro di testo

Tripling time A quantity increases according to the exponential function y(t) = y₀eᵏᵗ. What is the tripling time for the quantity? What is the time required for the quantity to increase p-fold?

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