6. The solution for the following problem is incorrect. State what is wrong and then correctly evaluate the integral. In r dr vdu uv u = 23 dv = Inr dr -31 dr du = 3r dr 1 v = - %3D 3x dr %3D
6. The solution for the following problem is incorrect. State what is wrong and then correctly evaluate the integral. In r dr vdu uv u = 23 dv = Inr dr -31 dr du = 3r dr 1 v = - %3D 3x dr %3D
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...
Related questions
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.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd667458b-01a8-490f-ac98-ae7bd5417420%2F4e33db79-01a0-431f-b324-83ed394abe31%2Fy011xln_processed.jpeg&w=3840&q=75)
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 now
This is a popular solution!
Step by step
Solved 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.Recommended textbooks for you
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:
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