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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 7.3.38
Chapter 7, Problem 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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Key Concepts

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

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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Related Practice
Textbook Question

10–19. Derivatives Find the derivatives of the following functions.

f(x) = ln(3 sin² 4x)

Textbook Question

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)

Textbook Question

37–56. Integrals Evaluate each integral.

∫ dx/x√(16 + x²)

Textbook Question

27–28. Curve sketching Use the graphing techniques of Section 4.4 to graph the following functions on their domains. Identify local extreme points, inflection points, concavity, and end behavior. Use a graphing utility only to check your work.

f(x) = ln x – ln² x

Textbook Question

37–56. Integrals Evaluate each integral.

∫ tanh²x dx (Hint: Use an identity.)

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Textbook Question

10–19. Derivatives Find the derivatives of the following functions.

f(x) = (sinh x) / (1 + sinh x)