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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![## Integration Problem
### Problem Statement
**6. Integrate. Show your work.**
*Hint: partial fraction decomposition*
\[ \int \frac{2x^2 - x + 4}{x(x^2 + 4)} \, dx \]
### Solution Approach
To integrate this function, we can use the method of partial fraction decomposition. Partial fraction decomposition is a technique used to break down a complex fraction into simpler fractions that are easier to integrate.
### Detailed Steps
1. **Factor the Denominator (if possible):**
The denominator of our integrand is \( x(x^2 + 4) \), which is already factored.
2. **Set Up Partial Fractions:**
The integrand can be expressed as a sum of simpler fractions:
\[ \frac{2x^2 - x + 4}{x(x^2 + 4)} = \frac{A}{x} + \frac{Bx + C}{x^2 + 4} \]
where \( A \), \( B \), and \( C \) are constants to be determined.
3. **Determine Constants \( A \), \( B \), and \( C \):**
Multiply both sides by the common denominator \( x(x^2 + 4) \) to eliminate the denominators:
\[ 2x^2 - x + 4 = A(x^2 + 4) + (Bx + C)x \]
Expand and collect like terms:
\[ 2x^2 - x + 4 = Ax^2 + 4A + Bx^2 + Cx \]
\[ 2x^2 - x + 4 = (A + B)x^2 + Cx + 4A \]
By comparing coefficients, we can solve for \( A \), \( B \), and \( C \):
Comparing \( x^2 \) terms: \( A + B = 2 \)
Comparing \( x \) terms: \( C = -1 \)
Comparing constant terms: \( 4A = 4 \) gives \( A = 1 \)
Substitute \( A = 1 \) back into \( A + B = 2 \):
\( 1 + B = 2 \) gives \( B](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3bf7da54-5138-4b57-a0f9-aec30fa8e8cf%2Fc61271a0-cd7b-4472-b2d6-36c68482401f%2F4bovo58.jpeg&w=3840&q=75)
Transcribed Image Text:## Integration Problem
### Problem Statement
**6. Integrate. Show your work.**
*Hint: partial fraction decomposition*
\[ \int \frac{2x^2 - x + 4}{x(x^2 + 4)} \, dx \]
### Solution Approach
To integrate this function, we can use the method of partial fraction decomposition. Partial fraction decomposition is a technique used to break down a complex fraction into simpler fractions that are easier to integrate.
### Detailed Steps
1. **Factor the Denominator (if possible):**
The denominator of our integrand is \( x(x^2 + 4) \), which is already factored.
2. **Set Up Partial Fractions:**
The integrand can be expressed as a sum of simpler fractions:
\[ \frac{2x^2 - x + 4}{x(x^2 + 4)} = \frac{A}{x} + \frac{Bx + C}{x^2 + 4} \]
where \( A \), \( B \), and \( C \) are constants to be determined.
3. **Determine Constants \( A \), \( B \), and \( C \):**
Multiply both sides by the common denominator \( x(x^2 + 4) \) to eliminate the denominators:
\[ 2x^2 - x + 4 = A(x^2 + 4) + (Bx + C)x \]
Expand and collect like terms:
\[ 2x^2 - x + 4 = Ax^2 + 4A + Bx^2 + Cx \]
\[ 2x^2 - x + 4 = (A + B)x^2 + Cx + 4A \]
By comparing coefficients, we can solve for \( A \), \( B \), and \( C \):
Comparing \( x^2 \) terms: \( A + B = 2 \)
Comparing \( x \) terms: \( C = -1 \)
Comparing constant terms: \( 4A = 4 \) gives \( A = 1 \)
Substitute \( A = 1 \) back into \( A + B = 2 \):
\( 1 + B = 2 \) gives \( B
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