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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![### Problem 4: Method of Partial Fractions
Use the method of partial fractions to evaluate the following integral:
\[
\int \frac{5x^2 - 3x + 2}{x^3 - 2x^2} \, dx.
\]
### Solution Steps
1. **Factor the Denominator**:
First, factor the denominator \( x^3 - 2x^2 \).
\[
x^3 - 2x^2 = x^2(x - 2)
\]
2. **Set Up Partial Fractions**:
Express the integrand as a sum of partial fractions.
\[
\frac{5x^2 - 3x + 2}{x^2(x - 2)} = \frac{A}{x} + \frac{B}{x^2} + \frac{C}{x - 2}
\]
3. **Solve for \( A \), \( B \), and \( C \)**:
Multiply through by the common denominator \( x^2(x - 2) \) to obtain:
\[
5x^2 - 3x + 2 = A x(x - 2) + B(x - 2) + C x^2
\]
4. **Integrate Each Term**:
Finally, integrate each partial fraction separately.
For detailed steps, refer to the educational content sections on partial fractions and integration techniques.
Additionally, graphical representations (if any such as graphs or diagrams) can be drawn to better illustrate the method. In this case, there are no specific diagrams provided, but conceptual diagrams showing partial fraction decomposition could be helpful.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fcf5ca60f-6906-44bf-a492-477ae53fc791%2Ff7a1f802-7837-4089-a559-8181bdfc6694%2Fcq2sci8_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Problem 4: Method of Partial Fractions
Use the method of partial fractions to evaluate the following integral:
\[
\int \frac{5x^2 - 3x + 2}{x^3 - 2x^2} \, dx.
\]
### Solution Steps
1. **Factor the Denominator**:
First, factor the denominator \( x^3 - 2x^2 \).
\[
x^3 - 2x^2 = x^2(x - 2)
\]
2. **Set Up Partial Fractions**:
Express the integrand as a sum of partial fractions.
\[
\frac{5x^2 - 3x + 2}{x^2(x - 2)} = \frac{A}{x} + \frac{B}{x^2} + \frac{C}{x - 2}
\]
3. **Solve for \( A \), \( B \), and \( C \)**:
Multiply through by the common denominator \( x^2(x - 2) \) to obtain:
\[
5x^2 - 3x + 2 = A x(x - 2) + B(x - 2) + C x^2
\]
4. **Integrate Each Term**:
Finally, integrate each partial fraction separately.
For detailed steps, refer to the educational content sections on partial fractions and integration techniques.
Additionally, graphical representations (if any such as graphs or diagrams) can be drawn to better illustrate the method. In this case, there are no specific diagrams provided, but conceptual diagrams showing partial fraction decomposition could be helpful.
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