Problem 18 - Calculus Early Transcendentals

Question

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

Step-by-Step Solution

Answer

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 1: Perform the first substitution

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

Step 2: Perform the second substitution

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

Step 3: Evaluate the integral

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

Step 4: Convert back to original variable x

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