Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Rate of Change
The relation between two quantities which displays how much greater one quantity is than another is called ratio.
Slope
The change in the vertical distances is known as the rise and the change in the horizontal distances is known as the run. So, the rise divided by run is nothing but a slope value. It is calculated with simple algebraic equations as:
Question
![**Question 8:**
For what values of \( p \) does the integral
\[ \int_{0}^{1} x^p \log_e x \, dx \]
converge?
---
In this problem, we are examining the convergence properties of the integral
\[ \int_{0}^{1} x^p \log_e x \, dx. \]
The integral will converge for certain values of the parameter \( p \).
To determine the values of \( p \) for which the integral converges, consider the behavior of the integrand \( x^p \log_e x \) as \( x \) approaches the limits of integration, especially near \( x = 0 \) where the integrand can potentially exhibit improper behavior.
The key is to analyze the bounds and determine under what conditions on \( p \), the integral remains finite.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6f0add4a-216e-42a4-b745-6e48fd2843da%2F5691ab77-e73d-4b64-bb6c-466615babe08%2Fx5u5ev5_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Question 8:**
For what values of \( p \) does the integral
\[ \int_{0}^{1} x^p \log_e x \, dx \]
converge?
---
In this problem, we are examining the convergence properties of the integral
\[ \int_{0}^{1} x^p \log_e x \, dx. \]
The integral will converge for certain values of the parameter \( p \).
To determine the values of \( p \) for which the integral converges, consider the behavior of the integrand \( x^p \log_e x \) as \( x \) approaches the limits of integration, especially near \( x = 0 \) where the integrand can potentially exhibit improper behavior.
The key is to analyze the bounds and determine under what conditions on \( p \), the integral remains finite.
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