Chapter 5: Q. 32 (page 478)

Use limits of definite integrals to calculate each of the improper integrals in Exercises.
$\int_{1}^{\infty} \frac{x + 1}{x^{2}} d x$

Short Answer

Expert verified

The improper integral $\int_{1}^{\infty} \frac{x + 1}{x^{2}} d x$ diverges.

Step by step solution

Step 1. Given information.

The given integral is the following.

$\int_{1}^{\infty} \frac{x + 1}{x^{2}} d x$

Step 2. Value of integral.

Consider function $f (x) = \frac{x + 1}{x^{2}}$

localid="1648879879677" $f (x) = \frac{x + 1}{x^{2}} = \frac{x}{x^{2}} + \frac{1}{x^{2}} = \frac{1}{x} + \frac{1}{x^{2}} \frac{1}{x} + \frac{1}{x^{2}} > \frac{1}{x^{2}}$

In the interval $[1, \infty)$ we have localid="1648879786507" $\frac{x + 1}{x^{2}} > \frac{1}{x^{2}} .$

As we know that localid="1648884171467" $\int_{1}^{\infty} \frac{1}{x^{2}} d x$ is converse and integral of functions less than $\frac{1}{x^{2}}$ will be converse.

But $\frac{x + 1}{x^{2}} ≮ \frac{1}{x^{2}}$ so it cannot be converse.

so $\frac{1}{x} + \frac{1}{x^{2}}$ is diverse.

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.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Use limits of definite integrals to calculate each of the improper integrals in Exercises.
$\int_{1}^{\infty} \frac{x + 1}{x^{2}} d x$

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Value of integral.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

Calculus

Mechanics Maths

Decision Maths

Geometry

Logic and Functions

Study anywhere. Anytime. Across all devices.

Use limits of definite integrals to calculate each of the improper integrals in Exercises.∫1∞x+1x2dx

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Value of integral.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

Calculus

Mechanics Maths

Decision Maths

Geometry

Logic and Functions

Study anywhere. Anytime. Across all devices.

Use limits of definite integrals to calculate each of the improper integrals in Exercises.
$\int_{1}^{\infty} \frac{x + 1}{x^{2}} d x$