/*! 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 24 Determine whether the improper i... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Problem 24

Determine whether the improper integral diverges or converges. Evaluate the integral if it converges. $$ \int_{0}^{\infty} \frac{x^{3}}{\left(x^{2}+1\right)^{2}} d x $$

Short Answer

Expert verified
The improper integral diverges.

Step by step solution

01

Find a comparison function and check its integral

We find a function simpler than \( \frac{x^{3}}{\left(x^{2}+1\right)^{2}} \), which in this case is \( \frac{1}{x} \). We then integrate \( \frac{1}{x} \) from 1 to infinity. The resulting improper integral \( \int_{1}^{\infty} \frac{1}{x} dx \) diverges.
02

Perform the comparison test

We can show that \( 0 \leq \frac{x^{3}}{\left(x^{2}+1\right)^{2}} \leq \frac{1}{x} \) holds for all \( x \geq 0 \). Using the comparison test, since \( \int_{1}^{\infty} \frac{1}{x} dx \) diverges, thus \( \int_{0}^{\infty} \frac{x^{3}}{\left(x^{2}+1\right)^{2}} dx \) also diverges.
03

State the conclusion

The improper integral \( \int_{0}^{\infty} \frac{x^{3}}{\left(x^{2}+1\right)^{2}} dx \) diverges. There is no finite value to evaluate.

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. $$ \int \frac{\sin ^{2} x-\cos ^{2} x}{\cos x} d x $$

Determine all values of \(p\) for which the improper integral converges. $$ \int_{0}^{1} \frac{1}{x^{p}} d x $$

Use a computer algebra system to find the integral. Graph the antiderivatives for two different values of the constant of integration.$$ \int \sec ^{4}(1-x) \tan (1-x) d x $$

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

Find the integral. Use a computer algebra system to confirm your result. $$ \int \csc ^{4} \theta d \theta $$
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Statistics

Read Explanation

Calculus

Read Explanation

Pure Maths

Read Explanation

Discrete Mathematics

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.