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
![## Task:
**Verify the Identity:**
\[
\frac{1}{x^2 - 1} = \frac{1}{2} \left( \frac{1}{x-1} - \frac{1}{x+1} \right)
\]
**And use this equation to evaluate the integral:**
\[
\int_{2}^{5} \frac{4}{x^2 - 1} \, dx
\]
---
### Explanation:
1. **Verify the Identity:**
- Start with the expression \(\frac{1}{x^2 - 1}\).
- Factor \(x^2 - 1\) as \((x-1)(x+1)\).
- Use partial fraction decomposition to rewrite \(\frac{1}{x^2 - 1}\) as \(\frac{1}{x-1} - \frac{1}{x+1}\).
- Verify that multiplying by \(\frac{1}{2}\) gives the correct decomposition as shown in the equation.
2. **Evaluate the Integral:**
- Use the verified identity to simplify the integral \(\int_{2}^{5} \frac{4}{x^2 - 1} \, dx\).
- Substitute the decomposition \(\frac{4}{x^2 - 1} = 2\left( \frac{1}{x-1} - \frac{1}{x+1} \right)\).
- Integrate each part separately:
\[
2 \int_{2}^{5} \left( \frac{1}{x-1} - \frac{1}{x+1} \right) \, dx
\]
- Find the antiderivatives and evaluate them at the limits of integration.
Completing this task will help in understanding partial fraction decomposition and applying it to evaluate definite integrals.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fdd3494d7-16cb-4f96-a1a3-f69334a1e65b%2F1b75a07e-0069-4b44-8de5-a09e7dca39ee%2Faujls4i_processed.png&w=3840&q=75)
Transcribed Image Text:## Task:
**Verify the Identity:**
\[
\frac{1}{x^2 - 1} = \frac{1}{2} \left( \frac{1}{x-1} - \frac{1}{x+1} \right)
\]
**And use this equation to evaluate the integral:**
\[
\int_{2}^{5} \frac{4}{x^2 - 1} \, dx
\]
---
### Explanation:
1. **Verify the Identity:**
- Start with the expression \(\frac{1}{x^2 - 1}\).
- Factor \(x^2 - 1\) as \((x-1)(x+1)\).
- Use partial fraction decomposition to rewrite \(\frac{1}{x^2 - 1}\) as \(\frac{1}{x-1} - \frac{1}{x+1}\).
- Verify that multiplying by \(\frac{1}{2}\) gives the correct decomposition as shown in the equation.
2. **Evaluate the Integral:**
- Use the verified identity to simplify the integral \(\int_{2}^{5} \frac{4}{x^2 - 1} \, dx\).
- Substitute the decomposition \(\frac{4}{x^2 - 1} = 2\left( \frac{1}{x-1} - \frac{1}{x+1} \right)\).
- Integrate each part separately:
\[
2 \int_{2}^{5} \left( \frac{1}{x-1} - \frac{1}{x+1} \right) \, dx
\]
- Find the antiderivatives and evaluate them at the limits of integration.
Completing this task will help in understanding partial fraction decomposition and applying it to evaluate definite integrals.
Expert Solution
Step 1
Step by step
Solved in 4 steps with 4 images
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