Problem 25
Question
Determine whether each integral is convergent. If the integral is convergent, compute its value. $$ \int_{e}^{\infty} \frac{d x}{x \ln x} $$
Step-by-Step Solution
Verified Answer
The integral \( \int_{e}^{\infty} \frac{d x}{x \ln x} \) is divergent.
1Step 1: Understanding the Integral
We have the integral \( \int_{e}^{\infty} \frac{1}{x \ln x} \, dx \). This is an improper integral because the upper limit of integration is infinity. We need to determine whether it is convergent or divergent.
2Step 2: Rewrite the Integral as a Limit
Express the integral in terms of a limit to handle the infinity: \( \int_{e}^{b} \frac{1}{x \ln x} \, dx \) and then take the limit as \( b \to \infty \). So, we have:\[\lim_{b \to \infty} \int_{e}^{b} \frac{1}{x \ln x} \, dx\]
3Step 3: Solve the Integral
Substituting \( u = \ln x \), then \( du = \frac{1}{x} \, dx \). The bounds for \( u \) become \( \ln e = 1 \) and \( \ln b = \ln b \) as \( x \) goes from \( e \) to \( b \). The integral becomes:\[\lim_{b \to \infty} \int_{1}^{\ln b} \frac{1}{u} \, du\]
4Step 4: Evaluate the New Integral
The integral \( \int \frac{1}{u} \, du \) is \( \ln |u| + C \). Therefore, with the new bounds, we have:\[\lim_{b \to \infty} \left[ \ln |\ln b| - \ln |1| \right]\]
5Step 5: Compute the Limit
Since \( \ln |1| = 0 \), the expression simplifies to:\[\lim_{b \to \infty} \ln |\ln b| = \infty\]The limit diverges to infinity, indicating that the original integral is divergent.
Key Concepts
Convergence and DivergenceIntegration TechniquesLimits and Infinity
Convergence and Divergence
Improper integrals can either converge or diverge. To determine convergence, we need to check if the integral approaches a finite value as the bounds extend. If the integral reaches a finite number, it converges. If it goes to infinity, it diverges.
In our exercise, the integral \[ \int_{e}^{\infty} \frac{dx}{x \ln x} \]has an upper limit of infinity, making it improper. To find convergence or divergence, we analyze if \[ \lim_{b \to \infty} \int_{e}^{b} \frac{1}{x \ln x} \, dx \]results in a finite number. In this case, the integral \[ \int \frac{1}{u} \, du \]diverges as the limit extends to infinity \( \lim_{b \to \infty} \ln |\ln b| = \infty \). Thus, the original integral is divergent as it does not reach a finite number.
In our exercise, the integral \[ \int_{e}^{\infty} \frac{dx}{x \ln x} \]has an upper limit of infinity, making it improper. To find convergence or divergence, we analyze if \[ \lim_{b \to \infty} \int_{e}^{b} \frac{1}{x \ln x} \, dx \]results in a finite number. In this case, the integral \[ \int \frac{1}{u} \, du \]diverges as the limit extends to infinity \( \lim_{b \to \infty} \ln |\ln b| = \infty \). Thus, the original integral is divergent as it does not reach a finite number.
Integration Techniques
In solving improper integrals, employing integration techniques can simplify the task. A common method is to use substitution, which helps in transforming the integrand into a more manageable form.
For the integral\[ \int_{e}^{\infty} \frac{1}{x \ln x} \, dx \]substituting \( u = \ln x \) transforms the problem as follows:- Differential change: \( du = \frac{1}{x} \, dx \).- Integrand conversion: \( \frac{1}{x \ln x} \, dx = \frac{1}{u} \, du \).
After substitution, the integral becomes \[ \int_{1}^{\ln b} \frac{1}{u} \, du \],which represents the natural logarithm function \( \ln|u| + C \). Changing variables like this simplifies finding the convergence or divergence of an integral.
For the integral\[ \int_{e}^{\infty} \frac{1}{x \ln x} \, dx \]substituting \( u = \ln x \) transforms the problem as follows:- Differential change: \( du = \frac{1}{x} \, dx \).- Integrand conversion: \( \frac{1}{x \ln x} \, dx = \frac{1}{u} \, du \).
After substitution, the integral becomes \[ \int_{1}^{\ln b} \frac{1}{u} \, du \],which represents the natural logarithm function \( \ln|u| + C \). Changing variables like this simplifies finding the convergence or divergence of an integral.
Limits and Infinity
When dealing with improper integrals, limits and infinity play a significant role. The idea is to approach the problem by transforming infinity into a manageable limit problem.
In our exercise, the integral \[ \int_{e}^{\infty} \frac{1}{x \ln x} \, dx \]is expressed with a finite upper limit \( b \), and then you compute the limit \( \lim_{b \to \infty} \).
Understanding what happens as \( b \to \infty \) is crucial.- Solve the integral \( \int_{e}^{b} \frac{1}{x \ln x} \, dx \).- Evaluate the limit \( \lim_{b \to \infty} \ln |\ln b| \).
Since \( \ln |\ln b| \) tends toward infinity, the integral diverges. This illustrates how limits help us understand infinity's effect on integrals.
In our exercise, the integral \[ \int_{e}^{\infty} \frac{1}{x \ln x} \, dx \]is expressed with a finite upper limit \( b \), and then you compute the limit \( \lim_{b \to \infty} \).
Understanding what happens as \( b \to \infty \) is crucial.- Solve the integral \( \int_{e}^{b} \frac{1}{x \ln x} \, dx \).- Evaluate the limit \( \lim_{b \to \infty} \ln |\ln b| \).
Since \( \ln |\ln b| \) tends toward infinity, the integral diverges. This illustrates how limits help us understand infinity's effect on integrals.
Other exercises in this chapter
Problem 25
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