Problem 40 - Essential Calculus

Question

Determine whether the improper integral diverges or converges. Evaluate the integral if it converges, and check your results with the results obtained by using the integration capabilities of a graphing utility. $$ \int_{1}^{\infty} \frac{1}{x \ln x} d x $$

Step-by-Step Solution

Answer

The integral diverges because the limiting value does not exist.

Step 1: Setting up the limit

Because the integral is improper (the upper limit is infinity), we change this to a limit problem. Let \( b\) be a number greater than 1. We will evaluate the limit as \( b\) approaches infinity for this integral: \( \lim_{b\rightarrow \infty}\int_{1}^{b} \frac{1}{x \ln x} dx \)

Step 2: Integration by substitution

We can simplify the integral by using substitution. Let \( u = \ln x \). Then, \( du = \frac{1}{x} dx \). The integral is now: \( \lim_{b\rightarrow \infty}\int_{1}^{b} \frac{du}{u} \)

Step 3: Evaluate the Integral

The integral \(\int \frac{du}{u}\) equals \(\ln |u|\). Therefore, \(\lim_{b\rightarrow \infty}\int_{1}^{b} \frac{du}{u} = \lim_{b\rightarrow \infty} \ln |u| = \lim_{b\rightarrow \infty} \ln |\ln x|\). We then evaluate this from 1 to \(b\), which gives \( \lim_{b\rightarrow \infty} (\ln |\ln b| - \ln |\ln 1|) \)

Step 4: Evaluate the limit

\(\ln 1 = 0\), so that leaves us with \( \lim_{b\rightarrow \infty} \ln (\ln b) \). As b approaches infinity, \(\ln b\) also approaches infinity and therefore \(\ln (\ln b)\) approaches infinity. Therefore, the limit does not exist.