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Chapter 6: Problem 18

Evaluate the following integrals. Include absolute values only when needed. $$\int \frac{d x}{x \ln x \ln (\ln x)}$$

Short Answer

Expert verified
Based on the solution above, create a short answer question: Question: Evaluate the improper integral \(\int \frac{d x}{x \ln x \ln (\ln x)}\). Answer: \(\ln{\left| \ln (\ln x) \right|} + C\)

Step by step solution

01

Perform the first substitution

Let \(u = \ln x\), then \(du = \frac{1}{x}dx\). Substitute in the integral: $$\int \frac{d x}{x \ln x \ln (\ln x)} = \int \frac{du}{u \ln u}$$
02

Perform the second substitution

Let \(v = \ln u\), then \(dv = \frac{1}{u}du\). Substitute in the integral from step 1: $$\int \frac{du}{u \ln u} = \int \frac{dv}{v}$$
03

Evaluate the integral

The integral becomes a simple natural logarithm integral: $$\int \frac{dv}{v} = \ln{\left| v \right|} + C = \ln{\left| \ln u \right|} + C$$
04

Convert back to original variable x

\(\ln{\left| \ln u \right|} + C = \ln{\left| \ln (\ln x) \right|} + C\). Therefore, the integral evaluates to: $$\int \frac{d x}{x \ln x \ln (\ln x)} = \ln{\left| \ln (\ln x) \right|} + C$$

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