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Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 7 - Logarithmic, Exponential Functions, and Hyperbolic FunctionsProblem 7.3.108b
Chapter 7, Problem 7.3.108b

"Integral formula Carry out the following steps to derive the formula ∫ csch x dx = ln |tanh(x / 2)| + C (Theorem 7.6).


b. Use the identity for sinh(2u) to show that 2 / sinh(2u) = sech² u / tanh u."

Verified step by step guidance
1
Recall the double-angle identity for hyperbolic sine: \(\sinh(2u) = 2 \sinh u \cosh u\).
Start with the expression \(\frac{2}{\sinh(2u)}\) and substitute the identity: \(\frac{2}{2 \sinh u \cosh u} = \frac{1}{\sinh u \cosh u}\).
Express \(\frac{1}{\sinh u \cosh u}\) in terms of \(\tanh u\) and \(\operatorname{sech} u\) by rewriting the denominator: \(\sinh u = \frac{\tanh u}{\operatorname{sech} u}\) and \(\cosh u = \frac{1}{\operatorname{sech} u}\).
Use the definitions \(\tanh u = \frac{\sinh u}{\cosh u}\) and \(\operatorname{sech} u = \frac{1}{\cosh u}\) to rewrite the expression \(\frac{1}{\sinh u \cosh u}\) as \(\frac{\operatorname{sech}^2 u}{\tanh u}\).
Conclude that \(\frac{2}{\sinh(2u)} = \frac{\operatorname{sech}^2 u}{\tanh u}\), as required.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Hyperbolic Functions and Their Identities

Hyperbolic functions like sinh, cosh, and tanh are analogs of trigonometric functions but for a hyperbola. Key identities, such as sinh(2u) = 2 sinh u cosh u, help simplify expressions and are essential for manipulating integrals involving hyperbolic functions.
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Verifying Trig Equations as Identities

Integration of Hyperbolic Functions

Integrating hyperbolic functions often involves substitution and using their identities to rewrite the integrand. For example, integrating csch x requires expressing it in terms of sinh x and applying logarithmic integration techniques to arrive at the formula ∫ csch x dx = ln |tanh(x/2)| + C.
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Algebraic Manipulation of Hyperbolic Expressions

Proving identities like 2 / sinh(2u) = sech² u / tanh u requires careful algebraic manipulation using definitions: sech u = 1/cosh u and tanh u = sinh u / cosh u. Rearranging and substituting these expressions is crucial to verify the given identity.
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Related Practice
Textbook Question

Properties of exp(x) Use the inverse relations between ln x and exp(x), and the properties of ln x, to prove the following properties:


b. exp(x − y) = exp(x) / exp(y)

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Textbook Question

37–38. Caffeine After an individual drinks a beverage containing caffeine, the amount of caffeine in the bloodstream can be modeled by an exponential decay function, with a half-life that depends on several factors, including age and body weight. For the sake of simplicity, assume the caffeine in the following drinks immediately enters the bloodstream upon consumption.


An individual consumes two cups of coffee, each containing 90 mg of caffeine, two hours apart. Assume the half-life of caffeine for this individual is 5.7 hours.


b. Determine the amount of caffeine in the bloodstream 1 hour after drinking the second cup of coffee.

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Textbook Question

Theorem 7.8

Differentiate sinh⁻¹ x = ln (x + √(x² + 1)) to show that d/dx (sinh⁻¹ x) = 1 / √(x² + 1).

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Textbook Question

Overtaking City A has a current population of 500,000 people and grows at a rate of 3%/yr. City B has a current population of 300,000 and grows at a rate of 5%/yr.

b. Suppose City C has a current population of y₀ < 500,000 and a growth rate of p > 3%/yr. What is the relationship between y₀ and p such that Cities A and C have the same population in 10 years?

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Textbook Question

A running model A model for the startup of a runner in a short race results in the velocity function v(t) = a(1 - e⁻ᵗ/ᶜ), where a and c are positive constants, t is measured in seconds, and v has units of m/s. (Source: Joe Keller, A Theory of Competitive Running, Physics Today, 26, Sep 1973)


b. Using the velocity in part (a) and assuming s(0) = 0, find the position function s(t), for t ≥ 0.

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Textbook Question

Projection sensitivity

According to the 2014 national population projections published by the U.S. Census Bureau, the U.S. population is projected to be 334.4 million in 2020 with an estimated growth rate of 0.79%/yr.

b. Suppose the actual growth rate is instead 0.7%. What are the resulting doubling time and projected 2050 population?

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