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Magazine Chapter 8: Problem 12
Evaluate each improper integral or show that it diverges. \(\int_{e}^{\infty} \frac{\ln x}{x} d x\)
Short Answer
Expert verified
The integral \( \int_{e}^{\infty} \frac{\ln x}{x} \, dx \) diverges.
Step by step solution
01
Identify the Integral Type
We are given the improper integral \( \int_{e}^{\infty} \frac{\ln x}{x} \, dx \). It is improper because the upper limit of integration is infinity.
02
Rewrite the Integral with a Limit
To evaluate the improper integral, we express it as a limit: \( \lim_{b \to \infty} \int_{e}^{b} \frac{\ln x}{x} \, dx \).
03
Perform Integration
To integrate \( \frac{\ln x}{x} \), use the substitution method. Let \( u = \ln x \), so \( du = \frac{1}{x} \, dx \). Therefore, the integral becomes \( \int u \, du \), which results in \( \frac{u^2}{2} + C \). Substitute back to get \( \frac{(\ln x)^2}{2} \).
04
Evaluate the Definite Integral
Evaluate \( \left[ \frac{(\ln x)^2}{2} \right]_e^b = \frac{(\ln b)^2}{2} - \frac{(\ln e)^2}{2} \). Since \( \ln e = 1 \), the lower bound term simplifies to \( \frac{1}{2} \).
05
Evaluate the Limit
We now have \( \lim_{b \to \infty} \left( \frac{(\ln b)^2}{2} - \frac{1}{2} \right) \). As \( b \to \infty \), \( \ln b \to \infty \) and therefore \( \frac{(\ln b)^2}{2} \to \infty \).
06
Conclusion on Convergence
Since the expression tends toward infinity, the limit of the integral is also infinity. Thus, the integral diverges.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Integration Techniques
Integration is the process of finding the integral of a function, and there are multiple techniques available to achieve this. For the integral \( \int_{e}^{\infty} \frac{\ln x}{x} \, dx \), we need to apply specific strategies because it involves a natural logarithm function combined with a rational expression. One effective technique is the **substitution method**.
- Substitution simplifies the integration process by transforming the integrand into a more manageable form.
- In the given problem, setting \( u = \ln x \) neatly simplifies the function, as \( du = \frac{1}{x} \, dx \), making the integral \( \int u \, du \).
- By applying this, we can integrate to obtain \( \frac{u^2}{2} \).
Integral Convergence
Convergence in integrals refers to whether the total area under a curve approaches a finite value as the limits of integration approach infinity or a point of discontinuity. In the context of improper integrals, we consider whether our integral yields a finite result.
- An integral converges if the limit results in a finite number.
- Conversely, it diverges if the result tends towards infinity or does not settle on a number.
Calculus Education
Mastering calculus involves understanding foundational concepts like limits, derivatives, and integrals. Improper integrals, in particular, are a classic example often encountered in calculus education.
- They introduce students to integrating functions over unbounded intervals or regions with discontinuities.
- These types of problems help in developing problem-solving skills relevant to real-world applications such as physics and engineering.
- Providing clear explanations, as seen in the step-by-step solutions, is vital for students to follow complex mathematical reasoning.
Infinity Limits
Infinity limits describe the behavior of mathematical expressions as a variable approaches infinity. They are pivotal in evaluating improper integrals since these integrals often extend towards infinity.
- With the exercise \( \int_{e}^{\infty} \frac{\ln x}{x} \, dx \), evaluating the limit involves letting the upper boundary \( b \to \infty \).
- We observed that as \( b \to \infty \), \( \ln b \to \infty \) results in \( \left( \frac{(\ln b)^2}{2} \right) \to \infty \), leading to divergence.
