Problem 38 - Calculus Early Transcendentals: Pearson New International Edition

Question

Evaluate \(\int_{-1}^{1} \frac{1}{x \sqrt{-\ln |x|}} d x\) or show that it diverges.

Step-by-Step Solution

Answer

The integral diverges.

Step 1: Analyze the Given Integral

First, observe the integral: \( \int_{-1}^{1} \frac{1}{x \sqrt{-\ln |x|}} \, dx \). This is an improper integral due to the presence of \(-\ln |x|\), which is undefined at \(x = 0\). Also, the natural logarithm can approach infinity near \(|x| = 1\), making it necessary to check the behavior of the integral at these points.

Step 2: Separate the Integral

To handle the improper nature, separate the integral at the singular point. \[\int_{-1}^{1} \frac{1}{x \sqrt{-\ln |x|}} \, dx = \int_{-1}^{0} \frac{1}{x \sqrt{-\ln |x|}} \, dx + \int_{0}^{1} \frac{1}{x \sqrt{-\ln x}} \, dx\]. Now, check each part separately for convergence.

Step 3: Examine Behavior at Singularities

Focus on \(\int_{0}^{1} \frac{1}{x \sqrt{-\ln x}} \, dx\). As \(x \to 0^+\), \(-\ln x \to \infty\), making \(\sqrt{-\ln x}\) large and the integrand \(\frac{1}{x \sqrt{-\ln x}}\) bounded. However, near \(x = 1\), \(-\ln x \to 0\) causing \(\sqrt{-\ln x}\) to approach 0 and hence \(\frac{1}{x \sqrt{-\ln x}}\) tends towards infinity.

Step 4: Test for Divergence at Endpoint

Analyzing \(\int_{0}^{1} \frac{1}{x \sqrt{-\ln x}} \, dx\) near \(x = 1\), observe that the behavior mimics \(\int_{0}^{1} \frac{1}{x} \, dx\), which diverges. Therefore, due to the unbounded behavior of \(\frac{1}{x \sqrt{-\ln x}}\) as \(x\) approaches 1, the integral diverges.

Step 5: Conclude Divergence

Since \(\int_{0}^{1} \frac{1}{x \sqrt{-\ln x}} \, dx\) is divergent, and no finite part can counterbalance a divergent \(-1\) to \(0\) integral, the entire integral from \(-1\) to \(1\) diverges.

Key Concepts

Integral Divergence

When dealing with improper integrals, it's important to determine whether the integral is convergent or divergent. An improper integral is one that has an infinite limit, a discontinuity within its interval, or both. In this exercise, \[ \int_{-1}^{1} \frac{1}{x \sqrt{-\ln |x|}} \, dx \]becomes improper because of the nature of the log function, which is undefined at 0 and somewhat untamed near |x|=1, raising questions around its convergence.Improper integrals need special attention. When the function itself becomes unbounded or the area under the curve extends infinitely, the integral may diverge. In this case, analyzing the bounds and behavior of the function across the interval is crucial. By recognizing the point of singularities, specifically at 0 and where -\(\ln|x| \to 0\), we must evaluate each segment separately. When one part diverges, the entire integral is typically considered divergent.For students struggling with such concepts, it’s beneficial to understand that divergence in an improper integral means the accumulated area under the curve doesn't stabilize or settle towards a finite value.

Natural Logarithm Behavior

The natural logarithm has intriguing behavior that plays a crucial role in the analysis of many integrals, especially those involving \(\ln |x|\).In this particular case, \( - \ln |x| \)is undefined at \(x = 0\) and exhibits interesting behavior:

• As \( x \to 0^+ \), \( -\ln x \to \infty \).
• As \(x \to 1^-\), \( -\ln x \to 0 \).

These tendencies are vital, as they determine the form and behavior of \(\sqrt{-\ln |x|}\) in the integrand. The logarithm's behavior directly affects the integrand \(\frac{1}{x \sqrt{-\ln |x|}}\) causing it to approach infinity at different points, impacting convergence or divergence.Understanding these tendencies of natural logarithms can help in anticipating such points of divergence in integrals. Recognizing when and where \(-\ln |x|\)will become large or small can assist greatly in making sense of the integral's behavior over its specified interval.

Convergence Analysis

Convergence analysis is a key part of dealing with integrals, especially improper ones like \(\int_{-1}^{1} \frac{1}{x \sqrt{-\ln |x|}} \, dx\).To ascertain convergence or divergence, we separate the integral and examine individual segments. A typical approach includes:

• Breaking down the integral into \(\int_{-1}^{0}\) and \(\int_{0}^{1}\).
• Testing each segment for convergence separately.

For example, near \(x=1\), integration resembles \(\int_{0}^{1} \frac{1}{x} \, dx\) which diverges, demonstrating that this part alone can't hold a finite value.So in convergence analysis:

• Examining limits of the function within its integration bounds is crucial.
• Determining points where integrands go unbounded helps in concluding divergence.

In our exercise, the critical endpoint is when \(-\ln x\) reduces to zero,making the integrand \(\frac{1}{x \sqrt{-\ln x}}\) unbounded. So, understanding convergence analysis helps us decide if the integral's total area aligns closely or diverges.