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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![### Calculating Indefinite Integrals Using Partial Fraction Decomposition
#### Problem Statement:
Calculate the indefinite integral using Partial Fraction Decomposition.
\[
\int \frac{x}{(x-2)(x+1)^2} \, dx
\]
### Explanation:
#### Step-by-Step Approach to Partial Fraction Decomposition:
To solve the integral using partial fraction decomposition, follow these steps:
1. **Express the Integrand as Partial Fractions:**
Decompose the function into a sum of simpler fractions. The fraction \(\frac{x}{(x-2)(x+1)^2}\) can be decomposed as:
\[
\frac{x}{(x-2)(x+1)^2} = \frac{A}{x-2} + \frac{B}{x+1} + \frac{C}{(x+1)^2}
\]
2. **Determine Coefficients (A, B, C):**
Solve for constants \(A\), \(B\), and \(C\) by equating the numerator on both sides after multiplying through by the denominator.
3. **Integrate Each Term:**
Once the coefficients are known, integrate each term separately.
4. **Combine the Results:**
Put together the results of the individual integrals to get the final answer.
Refer to educational resources or work through the decomposition and integration steps for detailed procedures on finding \(A\), \(B\), and \(C\) and performing the integrations.
This method helps in transforming a complex rational function into simpler fractions that can be integrated easily.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe4498e0d-5915-421f-a7a7-3ad8a56e2358%2F63b23243-d850-46ed-b293-af0ad6b7862d%2Fyhxxtu_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Calculating Indefinite Integrals Using Partial Fraction Decomposition
#### Problem Statement:
Calculate the indefinite integral using Partial Fraction Decomposition.
\[
\int \frac{x}{(x-2)(x+1)^2} \, dx
\]
### Explanation:
#### Step-by-Step Approach to Partial Fraction Decomposition:
To solve the integral using partial fraction decomposition, follow these steps:
1. **Express the Integrand as Partial Fractions:**
Decompose the function into a sum of simpler fractions. The fraction \(\frac{x}{(x-2)(x+1)^2}\) can be decomposed as:
\[
\frac{x}{(x-2)(x+1)^2} = \frac{A}{x-2} + \frac{B}{x+1} + \frac{C}{(x+1)^2}
\]
2. **Determine Coefficients (A, B, C):**
Solve for constants \(A\), \(B\), and \(C\) by equating the numerator on both sides after multiplying through by the denominator.
3. **Integrate Each Term:**
Once the coefficients are known, integrate each term separately.
4. **Combine the Results:**
Put together the results of the individual integrals to get the final answer.
Refer to educational resources or work through the decomposition and integration steps for detailed procedures on finding \(A\), \(B\), and \(C\) and performing the integrations.
This method helps in transforming a complex rational function into simpler fractions that can be integrated easily.
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