Problem 12 - Calculus Early Transcendentals: Pearson New International Edition

Question

Evaluate each improper integral or show that it diverges. \(\int_{e}^{\infty} \frac{\ln x}{x} d x\)

Step-by-Step Solution

Answer

The integral diverges.

Step 1: Analyze the Improper Integral

The given integral is \( \int_{e}^{\infty} \frac{\ln x}{x} \, dx \). An improper integral must be analyzed because the upper limit of integration is infinite. So we will substitute the upper limit with a variable \( t \) and then determines the limit as \( t \to \infty \).

Step 2: Set Up the Limit Expression

We express the improper integral as a limit: \( \int_{e}^{\infty} \frac{\ln x}{x} \, dx = \lim_{t \to \infty} \int_{e}^{t} \frac{\ln x}{x} \, dx \).

Step 3: Compute the Integral

To evaluate \( \int_{e}^{t} \frac{\ln x}{x} \, dx \), we recognize that \( \frac{\ln x}{x} \) is the derivative of \( \ln^2 x / 2 \). Thus, \( \int \frac{\ln x}{x} \, dx = \frac{1}{2}(\ln x)^2 + C \).

Step 4: Evaluate the Definite Integral

Evaluate the definite integral from \( e \) to \( t \): \[\left[ \frac{1}{2} (\ln x)^2 \right]_{e}^{t} = \frac{1}{2} (\ln t)^2 - \frac{1}{2} (\ln e)^2.\] The value of \( \ln e \) is \( 1 \), so this simplifies to: \[\frac{1}{2} (\ln t)^2 - \frac{1}{2}.\]

Step 5: Take the Limit as t Approaches Infinity

Now take the limit as \( t \to \infty \): \[ \lim_{t \to \infty} \left( \frac{1}{2} (\ln t)^2 - \frac{1}{2} \right).\] As \( t \) becomes very large, \( (\ln t)^2 \) also grows indefinitely, meaning the expression \( \frac{1}{2} (\ln t)^2 \) diverges to infinity.

Step 6: Conclusion About Convergence or Divergence

Since the limit diverges to infinity, the initial improper integral \( \int_{e}^{\infty} \frac{\ln x}{x} \, dx \) also diverges.

Key Concepts

Integration Limits

Improper integrals often challenge learners due to their undefined limits, as they extend to infinity. In the integral \( \int_{e}^{\infty} \frac{\ln x}{x} \, dx \), the upper limit is infinity, presenting a classic case of an improper integral. To tackle such integrals:

• Substitute the infinite limit with a variable, usually denoted by \( t \).
• Then express the original integral as the limit of a more standard definite integral as \( t \rightarrow \infty \).

For this integral, the substitution and reformation process transpires as follows:\[\int_{e}^{\infty} \frac{\ln x}{x} \, dx = \lim_{t \to \infty} \int_{e}^{t} \frac{\ln x}{x} \, dx\]This approach transforms an open-ended problem into a frame we can calculate using the definite integral followed by evaluating the limit as \( t \) approaches infinity.

Convergence and Divergence

The core goal when dealing with improper integrals is determining whether they converge to a specific value or diverge, spiraling towards infinity. In step 5 of the original solution, we note that as \( t \) becomes very large, the term \( (\ln t)^2 \) grows unbounded.
It falls on the behavior of the integrand and that of the resulting expression after taking the limit to conclude if it converges or diverges.
Here, we see the entire expression \( \lim_{t \to \infty} \left( \frac{1}{2} (\ln t)^2 - \frac{1}{2} \right) \) clearly diverges since \( (\ln t)^2 \to \infty \). This divergence tells us that the original integral does not settle to a specific value but rather extends indefinitely, highlighting its divergence.

Natural Logarithm Integration

Integrating functions involving the natural logarithm function, \( \ln x \), can be unique and enriching. In our problem, we tackle a specific function involving \( \ln x \):

• Here, \( \frac{\ln x}{x} \) is integrated over the interval from \( e \) to \( t \).
• The clever insight is to recognize that \( \frac{\ln x}{x} \) is the derivative of \( \frac{1}{2} (\ln x)^2 \).
• Integrating, we obtain \( \int \frac{\ln x}{x} \, dx = \frac{1}{2}(\ln x)^2 + C \).

This formula allows easy computation of definite integrals involving \( \ln x \) by substitution and reformation, reducing complications in integration. While the integral itself may be straightforward, understanding its application in improper cases is essential for convergence analysis. This technique bridges raw integration and the broader concept of assessing the behavior of functions as limits approach infinity.