Problem 58 - Thomas Calculus

Question

In Exercises \(35-64\) , use integration, the Direct Comparison Test, or the Limit Comparison Test to test the integrals for convergence. If more than one method applies, use whatever method you prefer. $$ \int_{2}^{\infty} \frac{1}{\ln x} d x $$

Step-by-Step Solution

Answer

The integral \(\int_{2}^{\infty} \frac{1}{\ln x} dx\) diverges by the direct comparison test.

Step 1: Analyze the Integral and Choose a Method

We need to determine whether the improper integral \(\int_{2}^{\infty} \frac{1}{\ln x} \ dx\) converges or diverges. Given the form of the function, we can consider using a comparison test.

Step 2: Select a Comparison Function

For direct comparison, let's compare the given function \(\frac{1}{\ln x}\) with a simpler function that we know the behavior of. Consider \(\frac{1}{\ln x}\) as comparable to \(\frac{1}{x}\), which is known to diverge since \(\int_{a}^{\infty} \frac{1}{x} \ dx\) diverges for any \(a > 0\).

Step 3: Apply the Direct Comparison Test

The function \(\frac{1}{\ln x}\) can be compared to \(\frac{1}{x}\) for \(x \geq e\). Notice that \(\ln x < x\) for \(x \geq e\), implying \(\frac{1}{\ln x} > \frac{1}{x}\). Since \(\int_{e}^{\infty} \frac{1}{x} \ dx\) diverges, by the direct comparison test, \(\int_{e}^{\infty} \frac{1}{\ln x} \ dx\) also diverges.

Step 4: Conclude the Convergence Status of the Integral

Because \(\int_{e}^{\infty} \frac{1}{\ln x} \ dx\) diverges and \(\int_{2}^{e} \frac{1}{\ln x} \ dx\) is a finite number, we conclude that \(\int_{2}^{\infty} \frac{1}{\ln x} \ dx\) diverges overall.

Key Concepts

Improper Integrals

In calculus, improper integrals extend the concepts of definite integrals to functions that have infinite limits of integration or singularities along the interval. An integral is considered improper if it involves:

• The integrand having an infinite discontinuity within the range of integration.
• Either or both of the bounds being infinite.

The example integral, \( \int_{2}^{\infty} \frac{1}{\ln x} \ dx \), is improper because it extends to infinity. To find out if such integrals converge (approach a finite value) or diverge (grow without bound), specific tests are applied, such as comparison tests. These tests help evaluate the behavior of the integral without necessarily finding an explicit antiderivative.

Convergence Tests

Convergence tests are tools used to determine whether an improper integral converges or diverges. The essential idea is to compare the integral of interest with another function whose convergence we already know.

• If an improper integral converges, it means calculating the total area under the curve results in a finite value.
• If it diverges, the area grows indefinitely large, implying no finite limit exists.

Various tests like the Comparison Test, Limit Comparison Test, and Integral Test are used to determine convergence. In the given exercise, the Direct Comparison Test is utilized to examine convergence. It's crucial to systematically choose the applicable test to ascertain the nature of the integral efficiently.

Direct Comparison Test

The Direct Comparison Test is an effective method for determining the convergence or divergence of improper integrals. This test involves comparing the integral in question with a benchmark function that is easier to evaluate:

• If \( f(x) \leq g(x) \) for all \( x \) in the interval and \( \int g(x) \) converges, then \( \int f(x) \) also converges.
• Conversely, if \( f(x) \geq g(x) \) and \( \int g(x) \) diverges, then \( \int f(x) \) also diverges.

In our example, the comparison function \( \frac{1}{x} \) was chosen because its integral is known to diverge. Since \( \frac{1}{\ln x} > \frac{1}{x} \) for \( x \geq e \), by the Direct Comparison Test, the given integral \( \int_{2}^{\infty} \frac{1}{\ln x} \ dx \) diverges.

Integration Techniques

Calculating improper integrals often requires specific integration techniques or adjustments. While basic integration may suffice for straightforward integrals, improper integrals may need comparison or limit approaches.Several techniques are frequently used:

• Substitution can simplify the integrand.
• Parts integration, which can break down products or simplify when products appear.
• Comparison methods, as used here, imply evaluating the convergence behavior against known benchmarks.

In this problem, instead of finding an antiderivative for \( \frac{1}{\ln x} \), comparison with the known diverging function \( \frac{1}{x} \) provided an efficient way to determine the convergence of the improper integral. This illustrates how using logical reasoning and known results can simplify complex integration tasks.