Chapter 5: Q. 15 (page 477)

Express each improper integral in Exercises 15–20 as a sum of limits of proper definite integrals. Do not calculate any integrals or limits; just write them down.
$\int_{1}^{\infty} \frac{1}{x - 1} d x$

Short Answer

Expert verified

The integral is $\int_{1}^{\infty} \frac{1}{x - 1} d x = \lim_{A \to 1^{+}} \int_{A}^{2} \frac{1}{x - 1} d x + \lim_{B \to \infty} \int_{2}^{8} \frac{1}{x - 1} d x$ .

Step by step solution

Step 1. Given information.

We are given an integrals,

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

Step 2. Graph of function

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

The graph is as follows,

Step 3. Expressing the Integral.

It can be seen from the graph that the function has vertical asymptote at $x = 1$ .

That is split the integral at some point, $x = 2$ , in order to separately consider the limit as $x \to \infty$ ,

So,

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

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Express each improper integral in Exercises 15–20 as a sum of limits of proper definite integrals. Do not calculate any integrals or limits; just write them down.
$\int_{1}^{\infty} \frac{1}{x - 1} d x$

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Graph of function

Step 3. Expressing the Integral.

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Express each improper integral in Exercises 15–20 as a sum of limits of proper definite integrals. Do not calculate any integrals or limits; just write them down.∫1∞1x-1dx

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

Logic and Functions

Geometry

Calculus

Discrete Mathematics

Probability and Statistics

Pure Maths

Study anywhere. Anytime. Across all devices.

Express each improper integral in Exercises 15–20 as a sum of limits of proper definite integrals. Do not calculate any integrals or limits; just write them down.
$\int_{1}^{\infty} \frac{1}{x - 1} d x$