Chapter 5: Q. 46 (page 478)

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

Short Answer

Expert verified

The value is $\infty$ .

Step by step solution

Step 1. Given nformation.

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

Step 2. Value of the limit.

Calculating the limit,

$\int_{1}^{\infty} \frac{\ln x}{x} d x = \lim_{t \to \infty} \int_{1}^{t} \ln x \cdot \frac{1}{x} d x Substitute \ln x = u \Rightarrow \frac{1}{x} d x = d u Then \int \ln x \cdot \frac{1}{x} d x = \int u \cdot d u = \frac{u^{2}}{2} = \frac{1}{2} (\ln x)^{2} T h e r e f o r e e q u a t i o n b e c o m e s \int_{1}^{\infty} \frac{\ln x}{x} d x = \lim_{t \to \infty} \int_{1}^{t} \ln x \frac{1}{x} d x = \lim_{t \to \infty} {[\frac{1}{2} (\ln x)^{2}]}_{1}^{t} = \lim_{t \to \infty} \frac{1}{2} [(\ln t)^{2} - (\ln 1)^{2}] = \frac{1}{2} \lim_{t \to \infty} (\ln t)^{2} = \infty$

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

Short Answer

Step by step solution

Step 1. Given nformation.

Step 2. Value of the limit.

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

Short Answer

Step by step solution

Step 1. Given nformation.

Step 2. Value of the limit.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Logic and Functions

Theoretical and Mathematical Physics

Calculus

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Applied Mathematics

Discrete Mathematics

Study anywhere. Anytime. Across all devices.

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