/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 60 Integral proof Prove the formula... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Problem 60

Integral proof Prove the formula \(\int\) coth \(x \, d x=\ln |\sinh x|+C\) of Theorem 7.6.

Short Answer

Expert verified
Question: Prove that the integral of the function coth(x) is equal to ln|sinh(x)| + C. Answer: To prove this, we first rewrite coth(x) as a ratio of hyperbolic functions: coth(x) = cosh(x) / sinh(x). Then, we apply the substitution method, where u = sinh(x) and du/dx = cosh(x). After substituting and solving the integral, we find that the integral of coth(x) equals ln|sinh(x)| + C.

Step by step solution

01

Rewrite the function coth(x) as a ratio of hyperbolic functions

Recall that \(\text{coth}(x) = \frac{\text{cosh}(x)}{\text{sinh}(x)}\). Our integral becomes: $$ \int \text{coth}(x) \, dx = \int \frac{\text{cosh}(x)}{\text{sinh}(x)} dx$$
02

Apply the substitution method

Let \(u=\text{sinh}(x)\), then \(\frac{du}{dx}=\text{cosh}(x)\). Rearrange for \(dx\): $$dx = \frac{du}{\text{cosh}(x)}$$ Now substitute in our integral: $$\int \frac{\text{cosh}(x)}{\text{sinh}(x)} dx = \int \frac{1}{u} du$$
03

Solve the integral

The integral of \(\frac{1}{u}\) is straightforward: $$\int \frac{1}{u} du = \ln|u| + C$$
04

Substitute back the original function

Recall that \(u = \text{sinh}(x)\), so we have: $$\ln |u| + C = \ln |\text{sinh}(x)| + C$$ Hence, we have proved the formula: $$\int \text{coth}(x) \, dx = \ln |\text{sinh}(x)| + C$$

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Hyperbolic Functions
Hyperbolic functions are mathematical functions that are analogs to trigonometric functions but for hyperbolas rather than circles. These functions are important in various fields such as calculus, physics, and engineering.
Here are the main hyperbolic functions:
  • Hyperbolic sine, \(\sinh(x)\), is defined by \(\sinh(x) = \frac{e^x - e^{-x}}{2}\).
  • Hyperbolic cosine, \(\cosh(x)\), is \(\cosh(x) = \frac{e^x + e^{-x}}{2}\).
  • The hyperbolic tangent, \(\tanh(x)\), combines them as \(\tanh(x) = \frac{\sinh(x)}{\cosh(x)}\).
  • Similarly, the hyperbolic cotangent, \(\coth(x)\), which is used in the original exercise, is given by \(\coth(x) = \frac{\cosh(x)}{\sinh(x)}\).
Just like their trigonometric cousins, hyperbolic functions have unique identities and properties that make them useful in integration and solving differential equations.
Understanding these functions and their relations gives students a strong foundation for tackling complex integrals and proofs, like the one in the exercise.
Substitution Method
The substitution method is a powerful technique in calculus used to simplify an integral by changing variables. It's akin to reverse chain rule for derivatives. The method replaces a complex expression with a single variable, making the integral easier to evaluate.
### The Substitution Process
  • Identify the part of the integral to substitute. In our problem, this was \(u = \sinh(x)\).
  • Differentiate this substitution with respect to the original variable, \(x\), resulting in \(\frac{du}{dx} = \cosh(x)\).
  • Express \(dx\) in terms of \(du\), yielding \(dx = \frac{du}{\cosh(x)}\).
  • Rewrite the integral using the new variables. Here, \(\int\coth(x) \, dx\) transformed into \(\int \frac{1}{u} \, du\).
The substitution method simplifies the problem, especially when direct integration is challenging. In this case, it converted a hyperbolic function based integral into a natural logarithm, which is straightforward.
Definite and Indefinite Integrals
Integrals in calculus can either be definite or indefinite, each serving distinct purposes.
### Indefinite Integrals
  • An indefinite integral, like in our exercise, is represented as \(\int f(x) \, dx = F(x) + C\), where \(C\) is an arbitrary constant.
  • It represents a family of functions and is equivalent to finding the antiderivative or original function before differentiation.
### Definite Integrals
  • Definite integrals compute a numerical value and are represented as \(\int_a^b f(x) \, dx\).
  • This process involves finding the area under the curve between two points \(a\) and \(b\) on a graph.
The solution of \(\int \coth(x) \, dx = \ln |\sinh(x)| + C\) demonstrates an indefinite integral. Understanding the distinction between definite and indefinite integrals helps students appreciate the comprehensive use of integration in mathematical analysis.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

L'H么pital loophole Explain why I'H么pital's Rule fails when applied to the limit \(\lim _{x \rightarrow \infty} \frac{\sinh x}{\cosh x}\) and then find the limit another way.

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 \(90 \mathrm{mg}\) of caffeine, two hours apart. Assume the half-life of caffeine for this individual is 5.7 hours. a. Determine the amount of caffeine in the bloodstream 1 hour after drinking the first cup of coffee. b. Determine the amount of caffeine in the bloodstream 1 hour after drinking the second cup of coffee.

Use a calculator to make a table similar to Table 7.1 to approximate the following limits. Confirm your result with l'H么pital's Rule. $$\lim _{x \rightarrow 0} \frac{\ln (1+x)}{x}$$

Points of intersection and area. a. Sketch the graphs of the functions \(f\) and \(g\) and find the \(x\) -coordinate of the points at which they intersect. b. Compute the area of the region described. \(f(x)=\sinh x, g(x)=\tanh x ;\) the region bounded by the graphs of \(f, g,\) and \(x=\ln 3\)

Compounded inflation The U.S. government reports the rate of inflation (as measured by the consumer price index) both monthly and annually. Suppose for a particular month, the monthly rate of inflation is reported as \(0.8 \% .\) Assuming this rate remains constant, what is the corresponding annual rate of inflation? Is the annual rate 12 times the monthly rate? Explain.
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Geometry

Read Explanation

Statistics

Read Explanation

Decision Maths

Read Explanation

Logic and Functions

Read Explanation

Theoretical and Mathematical Physics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.