Chapter 5: Q. 16 (page 477)

Express each improper integral in Exercises15–20 as a sum of limits of proper definite integrals. Do not calculate any integrals or limits; just write them down.
$\int_{1}^{\infty} \frac{1}{x - 2} d x$

Short Answer

Expert verified

The integral is $\int_{1}^{\infty} \frac{1}{x - 2} d x = \int_{1}^{3} \frac{1}{x - 2} d x + \lim_{A \to \infty} \int_{3}^{A} \frac{1}{x - 2} d x$ .

Step by step solution

Step 1. Given information.

We are given an integrals,

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

Step 2. Graph of function

Let $f (x) = \frac{1}{x - 2}$ ,

The graph is as follows,

Step 3. Expressing the Integral.

It can be seen from the graph that the function has vertical asymptote at $x = 2$ .

That is split the integral at some point, $x = 3$ , in order to separately consider the limit as $x \to \infty$

So,

\int_{1}^{\infty} \frac{1}{x - 2} d x = \int_{1}^{3} \frac{1}{x - 2} d x + \lim_{A \to \infty} \int_{3}^{A} \frac{1}{x - 2} d x

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.

Express each improper integral in Exercises15–20 as a sum of limits of proper definite integrals. Do not calculate any integrals or limits; just write them down.
$\int_{1}^{\infty} \frac{1}{x - 2} d x$

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Graph of function

Step 3. Expressing the Integral.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Decision Maths

Geometry

Probability and Statistics

Calculus

Logic and Functions

Discrete Mathematics

Study anywhere. Anytime. Across all devices.

Express each improper integral in Exercises15–20 as a sum of limits of proper definite integrals. Do not calculate any integrals or limits; just write them down.∫1∞1x-2dx

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Graph of function

Step 3. Expressing the Integral.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Decision Maths

Geometry

Probability and Statistics

Calculus

Logic and Functions

Discrete Mathematics

Study anywhere. Anytime. Across all devices.

Express each improper integral in Exercises15–20 as a sum of limits of proper definite integrals. Do not calculate any integrals or limits; just write them down.
$\int_{1}^{\infty} \frac{1}{x - 2} d x$