Question

Determine whether the following integrals converge or diverge. $$\int_{3}^{\infty} \frac{d x}{\ln x}(\text { Hint: } \ln x \leq x .)$$

Short answer

Answer: The improper integral \(\int_{3}^{\infty} \frac{dx}{\ln x}\) diverges.

Step-by-step solution

Identify the integral as improper

We can see that the given integral is improper because it has an infinite limit \(\infty\) in the upper bound. In order to determine if it converges or diverges, we will use the comparison test for integrals.

Use the comparison test

According to the comparison test for integrals, given two functions \(f(x)\) and \(g(x)\), if \(0 \leq f(x) \leq g(x)\) for all x in the interval \([a, \infty)\), then: - if \(\int_a^{\infty} g(x)dx\) converges, then \(\int_a^{\infty} f(x)dx\) also converges, - if \(\int_a^{\infty} f(x)dx\) diverges, then \(\int_a^{\infty} g(x)dx\) also diverges. We're given the hint that \(\ln x \leq x\), so we can use it to find a function \(g(x)\) to compare with the given integral.

Compare with a simpler function

Using the hint, we have: $$\frac{1}{\ln x} \geq \frac{1}{x}$$ Now we can set \(f(x) = \frac{1}{\ln x}\) and \(g(x) = \frac{1}{x}\). Then, our inequality becomes: $$0 \leq f(x) = \frac{1}{\ln x} \geq g(x) = \frac{1}{x}$$ for all \(x\) in the interval \([3, \infty)\).

Check if the integral of the simpler function converges or diverges

First, let's integrate the function \(g(x)\): $$\int_3^{\infty} \frac{1}{x} dx$$ To solve this improper integral, we rewrite it as a limit: $$\lim_{t\to\infty} \int_3^{t} \frac{1}{x} dx$$ Now, integrate \(g(x) = \frac{1}{x}\): $$\lim_{t\to\infty}\left[\ln|x|\right]_3^t$$ Now evaluate the limit: $$\lim_{t\to\infty} (\ln|t| - \ln|3|) = \infty$$ Since the integral of the simpler function \(g(x)\) diverges, we can conclude that our initial integral also diverges according to the comparison test.

Conclude the result

By applying the comparison test for integrals, using the hint provided, and detecting the divergence of the simpler function, we can conclude that the improper integral \(\int_{3}^{\infty} \frac{dx}{\ln x}\) diverges.

Key concepts

Comparison Test

The Comparison Test is a powerful tool used to determine whether an improper integral converges or diverges by comparing it with another integral whose behavior is already known.
It is especially helpful when working with functions that are difficult to integrate directly.

• We say we have two functions, \( f(x) \) and \( g(x) \).
• If \( g(x) \) is larger or equal to \( f(x) \) over a certain interval \([a, \infty)\), and the integral \( \int_a^{\infty} g(x) \, dx \) converges, then the integral \( \int_a^{\infty} f(x) \, dx \) also converges.
• Conversely, if \( \int_a^{\infty} f(x) \, dx \) diverges, then \( \int_a^{\infty} g(x) \, dx \) also diverges.

Utilizing this method requires identifying appropriate comparison functions and ensuring their relationships are valid over the specified interval.

Divergence

Divergence in the context of improper integrals refers to the situation where the integral does not settle to a finite number as one or both limits of integration approach infinity or extend to a discontinuity.
This concept helps us understand when an integral cannot be evaluated to a finite value.

• An integral \( \int_a^{\infty} f(x) \, dx \) is said to diverge if, as we take the limit of the upper bound to infinity, the integral keeps increasing without bound.
• In the case of \( \frac{1}{x} \), as seen in the solution, the indefinite integral leads to \( \lim_{t\to\infty} (\ln|t| - \ln|a|) = \infty \). This indicates divergence because the result tends towards infinity rather than stabilizing.

Recognizing divergence is crucial in determining the behavior of improper integrals, which assists in applying the comparison test accurately.

Limit of Integration

The Limit of Integration is a vital concept when dealing with improper integrals.
It highlights how the boundaries of integration—especially when they involve infinity—affect the nature of the solutions we obtain.

• Improper integrals arise when one or both limits extend to infinity, or when the function is undefined at a point within the limits.
• The process involves taking a limit to substitute the infinite bound—e.g., converting \( \int_a^{\infty} f(x) \, dx \) to \( \lim_{t\to \infty} \int_a^{t} f(x) \, dx \).
• This approach helps in handling the tension of infinite intervals, allowing us to examine whether integration results in finite or infinite values, which is crucial in determining convergence or divergence.

Understanding limits of integration is foundational for making well-informed decisions about the behavior of improper integrals.