Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN: 9781285741550
Author: James Stewart
Publisher: Cengage Learning
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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.
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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.
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