/*! 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 25 Determine whether the improper i... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Problem 25

Determine whether the improper integral diverges or converges. Evaluate the integral if it converges. $$ \int_{0}^{\infty} \frac{1}{e^{x}+e^{-x}} d x $$

Short Answer

Expert verified
The improper integral diverges.

Step by step solution

01

Understanding the function

The function to be integrated is \( \frac{1}{e^{x}+e^{-x}} \). For any real number x, the denominator of this fraction is always positive and hence the function is defined for all x.
02

Rewrite the function

Write \(e^{-x}\) as \(\frac{1}{e^x}\). Hence the function simplifies to a sum of two terms: \( \frac{1}{2e^{x}} + \frac{1}{2e^{-x}} \). By splitting the integral, the integral of the function from 0 to infinity becomes the sum of two integrals - one for each term. Thus, the original integral is equivalent to \( \frac{1}{2} \int_{0}^{\infty} \frac{1}{e^{x}} d x + \frac{1}{2} \int_{0}^{\infty} \frac{1}{e^{-x}} dx \).
03

Evaluating the first integral

We can evaluate the first integral: \( \frac{1}{2} \int_{0}^{\infty} \frac{1}{e^{x}} d x \). The antiderivative of \( \frac{1}{e^{x}}\) is \( -e^{-x} \), which gives us \( -\frac{1}{2} e^{-x} \) from 0 to infinity. Plugging in 0 and infinity for (x), we have \( \frac{1}{2} \).
04

Evaluating the second integral

Next, we evaluate the second integral: \( \frac{1}{2} \int_{0}^{\infty} \frac{1}{e^{-x}} dx \) = \( \frac{1}{2} \int_{0}^{\infty} e^{x} dx \). The antiderivative of \( e^{x}\) is \( e^{x} \), which gives us \( \frac{1}{2} e^{x} \) from 0 to infinity. Step 4 does not exist for an evaluation of infinity - this makes the integral diverge.
05

Determine convergence or divergence

Since the second integral \( \frac{1}{2} \int_{0}^{\infty} e^{x} dx \) diverges, the original integral also diverges.

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!

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

Find the integral. Use a computer algebra system to confirm your result. $$ \int \csc ^{2} 3 x \cot 3 x d x $$

Laplace Transforms Let \(f(t)\) be a function defined for all positive values of \(t\). The Laplace Transform of \(f(t)\) is defined by \(F(s)=\int_{0}^{\infty} e^{-s t} f(t) d t\) if the improper integral exists. Laplace Transforms are used to solve differential equations. Find the Laplace Transform of the function. $$ f(t)=\sin a t $$

(a) The improper integrals \(\int_{1}^{\infty} \frac{1}{x} d x \quad\) and \(\int_{1}^{\infty} \frac{1}{x^{2}} d x\) diverge and converge, respectively. Describe the essential differences between the integrands that cause one integral to converge and the other to diverge. (b) Sketch a graph of the function \(y=\sin x / x\) over the interval \((1, \infty)\). Use your knowledge of the definite integral to make an inference as to whether or not the integral \(\int_{1}^{\infty} \frac{\sin x}{x} d x\) converges. Give reasons for your answer. (c) Use one iteration of integration by parts on the integral in part (b) to determine its divergence or convergence.

Use a computer algebra system to evaluate the definite integral. In your own words, describe how you would integrate \(\int \sec ^{m} x \tan ^{n} x d x\) for each condition. (a) \(m\) is positive and even. (b) \(n\) is positive and odd. (c) \(n\) is positive and even, and there are no secant factors. (d) \(m\) is positive and odd, and there are no tangent factors.

Continuous Functions In Exercises 73 and \(74,\) find the value of \(c\) that makes the function continuous at \(x=0\). \(f(x)=\left\\{\begin{array}{ll}\frac{4 x-2 \sin 2 x}{2 x^{3}}, & x \neq 0 \\\ c, & x=0\end{array}\right.\)
See all solutions

Recommended explanations on Math Textbooks

Logic and Functions

Read Explanation

Pure Maths

Read Explanation

Mechanics Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Discrete Mathematics

Read Explanation

Statistics

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.