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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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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

Acceleration, velocity, position Suppose the acceleration of an object moving along a line is given by a(t) = -k v(t), where k is a positive constant and v is the object's velocity. Assume the initial velocity and position are given by v(0) = 10 and s(0) = 0, respectively.

c. Use the fact that dv/dt = (dv/ds)(ds/dt) (by the Chain Rule) to find the velocity as a function of position.

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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?

Textbook Question

Velocity of falling body Refer to Exercise 95, which gives the position function for a falling body. Use m = 75 kg and k = 0.2.


c. How long does it take for the BASE jumper to reach a speed of 45 m/s (roughly 100 mi/hr)?

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Textbook Question

Depreciation of equipment A large die-casting machine used to make automobile engine blocks is purchased for \$2.5 million. For tax purposes, the value of the machine can be depreciated by 6.8% of its current value each year.


b. After how many years is the value of the machine 10% of its original value?

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)).

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.