Chapter 5: Q. 48 (page 478)

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

Short Answer

Expert verified

The value is $\infty$ .

Step by step solution

Step 1. Given information.

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

Step 2. Value of the integral.

On simplifying,

$Let u = \ln (x) i.e d u = \frac{1}{x} d x \int \frac{1}{x \ln (x)} d x = \int \frac{1}{u} d u = \ln | u | T h e r e f o r e, \int_{1}^{\infty} \frac{1}{x \ln x} d x = \lim_{z \to 0} \int_{1}^{B} \frac{1}{x \ln x} d x = \lim_{B \to \infty} [\ln | \ln (x) |]_{1}^{B} = \frac{1}{2} \lim_{B \to \infty} [\ln | \ln (B) | - \ln | \ln (1) |] = \frac{1}{2} \lim_{B \to \infty} [\ln | \ln (B) | - \ln | 0 |] = \infty (D i v e r g e s)$

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

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Value of the integral.

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Use limits of definite integrals to calculate each of the improper integrals in Exercises 21–56.∫1∞1xlnxdx

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

Theoretical and Mathematical Physics

Geometry

Discrete Mathematics

Probability and Statistics

Calculus

Logic and Functions

Study anywhere. Anytime. Across all devices.

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