Skip to main content
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 7.53
Chapter 7, Problem 7.53

37–56. Integrals Evaluate each integral.
∫ (cosh z) / (sinh² z) dz

Verified step by step guidance
1
Recognize that the integral involves hyperbolic functions: \(\cosh z\) and \(\sinh z\). Recall the derivatives: \(\frac{d}{dz} \sinh z = \cosh z\) and \(\frac{d}{dz} \cosh z = \sinh z\).
Rewrite the integral as \(\int \frac{\cosh z}{\sinh^{2} z} \, dz = \int \cosh z \cdot \sinh^{-2} z \, dz\) to see the structure more clearly.
Use substitution by letting \(u = \sinh z\). Then, \(du = \cosh z \, dz\), which means \(\cosh z \, dz = du\).
Substitute into the integral to get \(\int u^{-2} \, du\), which simplifies the integral to a power function of \(u\).
Integrate \(\int u^{-2} \, du\) using the power rule for integrals, then substitute back \(u = \sinh z\) to express the answer in terms of \(z\).

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
1m
Was this helpful?

Key Concepts

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

Hyperbolic Functions

Hyperbolic functions, such as sinh(z) and cosh(z), are analogs of trigonometric functions but based on exponential functions. They satisfy identities like cosh²(z) - sinh²(z) = 1, which are useful in simplifying expressions and integrals involving these functions.
Recommended video:
5:50
Asymptotes of Hyperbolas

Integration Techniques for Rational Functions

Integrals involving ratios of functions often require substitution or rewriting the integrand to a simpler form. Recognizing derivatives within the integrand, such as identifying if the numerator is the derivative of the denominator, helps in applying substitution effectively.
Recommended video:
6:04
Intro to Rational Functions

Substitution Method in Integration

The substitution method involves changing variables to simplify an integral. By letting u equal a function inside the integral (e.g., u = sinh(z)), the integral can be transformed into a more straightforward form, making it easier to evaluate.
Recommended video:
07:33
Euler's Method
Related Practice
Textbook Question

7–28. Derivatives Evaluate the following derivatives.


d/dx (ln (cos² x))

Textbook Question

27–30. Designing exponential decay functions Devise an exponential decay function that fits the following data; then answer the accompanying questions. Be sure to identify the reference point (t = 0) and units of time.


Valium metabolism The drug Valium is eliminated from the bloodstream with a half-life of 36 hr. Suppose a patient receives an initial dose of 20 mg of Valium at midnight. How much Valium is in the patient’s blood at noon the next day? When will the Valium concentration reach 10% of its initial level?

Textbook Question

A calculator has a built-in sinh⁻¹ x function, but no csch⁻¹ x function. How do you evaluate csch⁻¹ 5 on such a calculator?

Textbook Question

15–20. Designing exponential growth functions Complete the following steps for the given situation.


a. Find the rate constant k and use it to devise an exponential growth function that fits the given data.

b. Answer the accompanying question.


Cell growth The number of cells in a tumor doubles every 6 weeks starting with 8 cells. After how many weeks does the tumor have 1500 cells?

Textbook Question

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?

Textbook Question

37–56. Integrals Evaluate each integral.

∫ sinh x / (1 + cosh x) dx

2
views