Chapter 5: Q. 47 (page 478)

Use limits of definite integrals to calculate each of the improper integrals in Exercises 21–56.
$\int_{0}^{1} \frac{1}{x \ln x} d x$

Short Answer

Expert verified

The value is $\infty (D i v e r g e n t)$ .

Step by step solution

Step 1. Given Information.

The integral is $\int_{0}^{1} \frac{1}{x \ln x} d x$ .

Step 2. Value of the integral.

On simplifying,

$\int_{0}^{1} \frac{1}{x \ln x} d x Let \ln x = u \frac{1}{x} = \frac{d u}{d x} d u = \frac{d x}{x} \int_{0}^{1} \frac{1}{x \ln x} d x = \int_{1}^{0} \frac{1}{x u} u d u = \int_{1}^{0} \frac{1}{u} d u = \log u]_{1}^{0} = \infty - 0 = \infty (Divergent)$

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 21–56.
$\int_{0}^{1} \frac{1}{x \ln x} d x$

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Value of the integral.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Geometry

Mechanics Maths

Logic and Functions

Calculus

Theoretical and Mathematical Physics

Applied Mathematics

Study anywhere. Anytime. Across all devices.

Use limits of definite integrals to calculate each of the improper integrals in Exercises 21–56.∫011xlnxdx

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Value of the integral.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Geometry

Mechanics Maths

Logic and Functions

Calculus

Theoretical and Mathematical Physics

Applied Mathematics

Study anywhere. Anytime. Across all devices.

Use limits of definite integrals to calculate each of the improper integrals in Exercises 21–56.
$\int_{0}^{1} \frac{1}{x \ln x} d x$