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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Question
Case 4
![### Educational Resource on Partial Fractions and Integration
#### Exercises and Explanations
6. **Question:** What are the forms (constant, linear, or quadratic) of the numerators in each fraction above?
7. **Question:** How does the form of the numerator compare with its denominator?
8. **Question:** Explain the differences in the form of the numerators for terms with irreducible quadratics (case 3) and repeating linear terms (case 2).
9. **Exercise:** Evaluate the integral.
\[
\int \frac{2x^2 - x + 4}{x^3 + 4x} \, dx
\]
#### Case 4: \( Q(x) \) Contains Repeated Irreducible Quadratic Factors
In this case, an example of a quadratic factor could be something like \( (x^2 + 2)^2 \) in the factorization of \( Q(x) \). We need to write a fraction for each power of \( x^2 + 2 \). For instance:
\[
\frac{8x^3 + 13x}{(x^2 + 2)^2} = \frac{Ax + B}{x^2 + 2} + \frac{Cx + D}{(x^2 + 2)^2}
\]
11. **Question:** How is the partial fraction decomposition in case 4 similar to cases 2 and 3?
12. **Exercise:** Write the partial fraction decomposition for the rational expression.
\[
\frac{3x^2 - 2x - 1}{x(x + 1)^2(x^2 + 4)^2}
\]
13. **Exercise:** Evaluate the integral.
\[
\int \frac{1 - x + 2x^2 - x^3}{x(x^2 + 1)^2} \, dx
\]
14. **Question:** Is partial fraction decomposition necessary for integrating all rational functions? How are the rational functions in problems 1, 2, and 3 different than the others?
15. **Task:** Using complete sentences, summarize the process of using partial fractions for integrating a rational function. Specify the algebraic tools needed throughout the process.
---
This section is intended to provide practice and deeper understanding of integrating rational functions using partial fraction decomposition. The examples and exercises illustrate](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F87082b8a-b4f5-455e-a232-6f60990eaa64%2Fbc1291a9-33f0-471f-aa02-a2507c9b0b77%2F5gvfxlr.jpeg&w=3840&q=75)
Transcribed Image Text:### Educational Resource on Partial Fractions and Integration
#### Exercises and Explanations
6. **Question:** What are the forms (constant, linear, or quadratic) of the numerators in each fraction above?
7. **Question:** How does the form of the numerator compare with its denominator?
8. **Question:** Explain the differences in the form of the numerators for terms with irreducible quadratics (case 3) and repeating linear terms (case 2).
9. **Exercise:** Evaluate the integral.
\[
\int \frac{2x^2 - x + 4}{x^3 + 4x} \, dx
\]
#### Case 4: \( Q(x) \) Contains Repeated Irreducible Quadratic Factors
In this case, an example of a quadratic factor could be something like \( (x^2 + 2)^2 \) in the factorization of \( Q(x) \). We need to write a fraction for each power of \( x^2 + 2 \). For instance:
\[
\frac{8x^3 + 13x}{(x^2 + 2)^2} = \frac{Ax + B}{x^2 + 2} + \frac{Cx + D}{(x^2 + 2)^2}
\]
11. **Question:** How is the partial fraction decomposition in case 4 similar to cases 2 and 3?
12. **Exercise:** Write the partial fraction decomposition for the rational expression.
\[
\frac{3x^2 - 2x - 1}{x(x + 1)^2(x^2 + 4)^2}
\]
13. **Exercise:** Evaluate the integral.
\[
\int \frac{1 - x + 2x^2 - x^3}{x(x^2 + 1)^2} \, dx
\]
14. **Question:** Is partial fraction decomposition necessary for integrating all rational functions? How are the rational functions in problems 1, 2, and 3 different than the others?
15. **Task:** Using complete sentences, summarize the process of using partial fractions for integrating a rational function. Specify the algebraic tools needed throughout the process.
---
This section is intended to provide practice and deeper understanding of integrating rational functions using partial fraction decomposition. The examples and exercises illustrate
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