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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 7.3.107b
Chapter 7, Problem 7.3.107b

Many formulas There are several ways to express the indefinite integral of sech x.


b. Show that ∫ sech x dx = sin⁻¹ (tanh x) + C. (Hint: Show that sech x = sech² x / √(1 − tanh² x) and then make a change of variables.)

Verified step by step guidance
1
Start with the integral you want to solve: \(\int \text{sech}\,x \, dx\).
Use the hint to rewrite \(\text{sech}\,x\) as \(\frac{\text{sech}^2 x}{\sqrt{1 - \tanh^2 x}}\). So the integral becomes \(\int \frac{\text{sech}^2 x}{\sqrt{1 - \tanh^2 x}} \, dx\).
Recognize that the derivative of \(\tanh x\) is \(\text{sech}^2 x\), which suggests the substitution \(u = \tanh x\). Then, \(du = \text{sech}^2 x \, dx\).
Rewrite the integral in terms of \(u\): it becomes \(\int \frac{1}{\sqrt{1 - u^2}} \, du\).
Recall that \(\int \frac{1}{\sqrt{1 - u^2}} \, du = \sin^{-1} u + C\). Substitute back \(u = \tanh x\) to get the final expression: \(\sin^{-1}(\tanh x) + C\).

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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 sech x and tanh x are analogs of trigonometric functions but based on exponential definitions. Understanding identities such as 1 - tanh² x = sech² x is crucial for manipulating expressions and simplifying integrals involving these functions.
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Verifying Trig Equations as Identities

Substitution Method in Integration

The substitution method involves changing variables to simplify an integral. By expressing the integral in terms of a new variable (e.g., u = tanh x), the integral becomes easier to evaluate, often transforming complicated expressions into standard forms.
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Euler's Method

Inverse Hyperbolic and Trigonometric Functions

Inverse functions like sin⁻¹ and tanh⁻¹ help express antiderivatives in closed form. Recognizing when an integral corresponds to an inverse trigonometric function, such as sin⁻¹(tanh x), allows for correct evaluation and interpretation of indefinite integrals.
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Derivatives of Other Inverse Trigonometric Functions
Related Practice
Textbook Question

Energy consumption On the first day of the year (t=0), a city uses electricity at a rate of 2000 MW. That rate is projected to increase at a rate of 1.3% per year.


b. Find the total energy (in MW-yr) used by the city over four full years beginning at t=0.

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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

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

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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Textbook Question

Definitions of hyperbolic sine and cosine Complete the following steps to prove that when the x- and y-coordinates of a point on the hyperbola x² - y² = 1 are defined as cosh t and sinh t, respectively, where t is twice the area of the shaded region in the figure, x and y can be expressed as

x = cosh t = (eᵗ + e⁻ᵗ) / 2 and y = sinh t = (eᵗ - e⁻ᵗ) / 2.



b. In Chapter 8, the formula for the integral in part (a) is derived:

∫ √(z² − 1) dz = (z/2)√(z² − 1) − (1/2) ln|z + √(z² − 1)| + C.

Evaluate this integral on the interval [1, x], explain why the absolute value can be dropped, and combine the result with part (a) to show that:

t = ln(x + √(x² − 1)).

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Textbook Question

Power lines A power line is attached at the same height to two utility poles that are separated by a distance of 100 ft; the power line follows the curve ƒ(x) = a cosh x/a. Use the following steps to find the value of a that produces a sag of 10 ft midway between the poles. Use a coordinate system that places the poles at x = ±50.

c. Use your answer in part (b) to find a, and then compute the length of the power line.

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