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Chapter 5: Q. 47 (page 478)

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

Short Answer

Expert verified

The value is $鈭� (D i v e r g e n t)$ .

Step by step solution

01

Step 1. Given Information.

The integral is ${鈭�}_{0}^{1} \frac{1}{x \ln x} d x$ .

02

Step 2. Value of the integral.

On simplifying,

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

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Most popular questions from this chapter

Solve the integral: $鈭� x e^{x} d x$

Solve the integral: $鈭� x \ln (x^{2}) d x$

Find three integrals in Exercises 27鈥�70 for which either algebra or u-substitution is a better strategy than integration by parts.

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

$鈭� \frac{1}{\sqrt{u}} d u$

Solve each of the integrals in Exercises 39鈥�74. Some integrals require trigonometric substitution, and some do not. Write your answers as algebraic functions whenever possible.

$鈭� \frac{\sqrt{9 - x^{2}}}{x} d x$

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