
Calculus: Early Transcendentals
8th Edition
ISBN: 9781285741550
Author: James Stewart
Publisher: Cengage Learning
expand_more
expand_more
format_list_bulleted
Question
![The image presents a calculus problem involving the integration of \(\frac{\ln x}{x^3}\, dx\). The issue at hand is identifying the error in the integral's solution and reevaluating it correctly.
1. **Problem Statement:**
- Evaluate the integral: \(\int \frac{\ln x}{x^3} \, dx\).
- The current solution is incorrect.
2. **Incorrect Solution:**
- The integration by parts formula used is:
\[
\int u \, dv = uv - \int v \, du
\]
- Substitution:
\[
u = x^3, \quad dv = \ln x \, dx
\]
\[
du = 3x^2 \, dx, \quad v = \frac{1}{x}
\]
3. **Steps in the Incorrect Solution:**
- Calculating \(uv\):
\[
x^3 \cdot \frac{1}{x} = x^2
\]
- Calculating \(\int v \, du\):
\[
\int \frac{1}{x} \cdot 3x^2 \, dx = \int 3x \, dx
\]
- Incorrect simplification:
\[
x^2 - \int 3x \, dx = x^2 - \frac{3}{2}x^2 + C
\]
\[
= -\frac{1}{2}x^2 + C
\]
4. **Observation on Errors:**
- The substitution choices for \(u\) and \(dv\) are incorrect, leading to a flawed integration by parts setup.
- Further algebraic manipulation errors are present in the simplification steps.
5. **Correct Evaluation (Guidance):**
- Reevaluating the integral requires selecting suitable \(u\) and \(dv\) to apply integration by parts accurately.
- Verify and recalculate each step meticulously.
This educational content highlights the importance of correctly applying integration techniques and thoroughly verifying solution steps. Users are encouraged to practice careful substitutions and simplifications to avoid common math errors.](https://content.bartleby.com/qna-images/question/d667458b-01a8-490f-ac98-ae7bd5417420/4e33db79-01a0-431f-b324-83ed394abe31/y011xln_thumbnail.jpeg)
Transcribed Image Text:The image presents a calculus problem involving the integration of \(\frac{\ln x}{x^3}\, dx\). The issue at hand is identifying the error in the integral's solution and reevaluating it correctly.
1. **Problem Statement:**
- Evaluate the integral: \(\int \frac{\ln x}{x^3} \, dx\).
- The current solution is incorrect.
2. **Incorrect Solution:**
- The integration by parts formula used is:
\[
\int u \, dv = uv - \int v \, du
\]
- Substitution:
\[
u = x^3, \quad dv = \ln x \, dx
\]
\[
du = 3x^2 \, dx, \quad v = \frac{1}{x}
\]
3. **Steps in the Incorrect Solution:**
- Calculating \(uv\):
\[
x^3 \cdot \frac{1}{x} = x^2
\]
- Calculating \(\int v \, du\):
\[
\int \frac{1}{x} \cdot 3x^2 \, dx = \int 3x \, dx
\]
- Incorrect simplification:
\[
x^2 - \int 3x \, dx = x^2 - \frac{3}{2}x^2 + C
\]
\[
= -\frac{1}{2}x^2 + C
\]
4. **Observation on Errors:**
- The substitution choices for \(u\) and \(dv\) are incorrect, leading to a flawed integration by parts setup.
- Further algebraic manipulation errors are present in the simplification steps.
5. **Correct Evaluation (Guidance):**
- Reevaluating the integral requires selecting suitable \(u\) and \(dv\) to apply integration by parts accurately.
- Verify and recalculate each step meticulously.
This educational content highlights the importance of correctly applying integration techniques and thoroughly verifying solution steps. Users are encouraged to practice careful substitutions and simplifications to avoid common math errors.
Expert Solution

This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution
Trending nowThis is a popular solution!
Step by stepSolved in 2 steps

Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, calculus and related others by exploring similar questions and additional content below.Similar questions
Recommended textbooks for you
- Calculus: Early TranscendentalsCalculusISBN:9781285741550Author:James StewartPublisher:Cengage LearningThomas' Calculus (14th Edition)CalculusISBN:9780134438986Author:Joel R. Hass, Christopher E. Heil, Maurice D. WeirPublisher:PEARSONCalculus: Early Transcendentals (3rd Edition)CalculusISBN:9780134763644Author:William L. Briggs, Lyle Cochran, Bernard Gillett, Eric SchulzPublisher:PEARSON
- Calculus: Early TranscendentalsCalculusISBN:9781319050740Author:Jon Rogawski, Colin Adams, Robert FranzosaPublisher:W. H. FreemanCalculus: Early Transcendental FunctionsCalculusISBN:9781337552516Author:Ron Larson, Bruce H. EdwardsPublisher:Cengage Learning

Calculus: Early Transcendentals
Calculus
ISBN:9781285741550
Author:James Stewart
Publisher:Cengage Learning

Thomas' Calculus (14th Edition)
Calculus
ISBN:9780134438986
Author:Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:PEARSON

Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:9780134763644
Author:William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:PEARSON

Calculus: Early Transcendentals
Calculus
ISBN:9781319050740
Author:Jon Rogawski, Colin Adams, Robert Franzosa
Publisher:W. H. Freeman


Calculus: Early Transcendental Functions
Calculus
ISBN:9781337552516
Author:Ron Larson, Bruce H. Edwards
Publisher:Cengage Learning