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Chapter 5: Q. 61 (page 478)

Determine the convergence or divergence of each improper integral in Exercises 57鈥�64 by comparing it to simpler improper integrals whose convergence or divergence is known or can be found directly.
${鈭�}_{1}^{鈭�} \frac{1 + \ln x}{x} d x$

Short Answer

Expert verified

The integral diverges.

Step by step solution

01

Step 1. Given information.

The given integral is ${鈭�}_{1}^{鈭�} \frac{1 + \ln x}{x} d x$ .

02

Step 2. Conclusion.

Now,

${鈭�}_{1}^{鈭�} \frac{1}{x} d x diverges on an unbounded interval [1, 鈭�) A l s o, \frac{1}{x} 鈮� \frac{1}{x} + \frac{\ln x}{x} for all x 鈭� [1, 鈭�) .$

Therefore,

${鈭�}_{1}^{鈭�} (\frac{1}{x} + \frac{\ln x}{x}) d x diverges on [1, 鈭�) since {鈭�}_{1}^{鈭�} \frac{1}{x} d x diverges.$

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Most popular questions from this chapter

Explain how to know when to use the trigonometric substitutions $x = a \sin 鈦� u, x = a \tan 鈦� u, a n d x = a \sec 鈦� u$ , Describe the trigonometric identity and the triangle that will be needed in each case. What are the possible values for $x$ and $u$ in each case?

Explain why, if $x = a s e c u$ , then $\sqrt{a^{2} s e c^{2} u 鈭� a^{2}}$ is $鈭� a \tan u$ if $x < 鈭� a$ and is $a \tan u$ if $x > a$ . Your explanation should include a discussion of domains and absolute values.

For each integral in Exercises 5鈥�8, write down three integrals that will have that form after a substitution of variables.

$鈭� \frac{1}{\sqrt{u}} d u$

Explain how to use long division to write the improper fraction $\frac{172}{5}$ as the sum of an integer and a proper fraction.

Solve the integral: $鈭� \ln (x^{3}) d x$ .

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