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12. Write the partial fraction decomposition for the rational expression. 3x2 - 2x- 1 x(x+ 1)2(x² + 4)2

12. Write the partial fraction decomposition for the rational expression. 3x2 - 2x- 1 x(x+ 1)2(x² + 4)2

Calculus: Early Transcendentals
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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Case 4

6. What are the forms (constant, linear, or quadratic) of the numerators in each fraction above? 7. How does the form of the numerator compare with its denominator? 8. Explain the differences in the form of the numerators for terms with irreducible quadratics (case 3) and repeating linear terms (case 2). 9. Evaluate the integral. 2x2-x+4 x3+ 4x xp Case 4. Q(x) Contains Repeated Irreducible Quadratic Factors Something like (x2 +2)2 occurs in the factorization of Q(x). In this case we need to write a fraction for each power of x2 + 2. For example, Cx + D (x² + 2)²* 8x3 +13x Ax + B (x2 + 2)2 %3D 11. How is the partial fraction decomposition in case 4 similar to case 2 and case 3? 12. Write the partial fraction decomposition for the rational expression. 3x2 - 2x-1 | x(x + 1)²(x² + 4)2 13. Evaluate the integral. EX-Z+xーI xp x(x² + 1)2 14. 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. Using compete sentences, summarize the process of using partial fractions for integrating a rational function. Specify the algebraic tools needed throughout the process.
Transcribed Image Text:6. What are the forms (constant, linear, or quadratic) of the numerators in each fraction above? 7. How does the form of the numerator compare with its denominator? 8. Explain the differences in the form of the numerators for terms with irreducible quadratics (case 3) and repeating linear terms (case 2). 9. Evaluate the integral. 2x2-x+4 x3+ 4x xp Case 4. Q(x) Contains Repeated Irreducible Quadratic Factors Something like (x2 +2)2 occurs in the factorization of Q(x). In this case we need to write a fraction for each power of x2 + 2. For example, Cx + D (x² + 2)²* 8x3 +13x Ax + B (x2 + 2)2 %3D 11. How is the partial fraction decomposition in case 4 similar to case 2 and case 3? 12. Write the partial fraction decomposition for the rational expression. 3x2 - 2x-1 | x(x + 1)²(x² + 4)2 13. Evaluate the integral. EX-Z+xーI xp x(x² + 1)2 14. 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. Using compete sentences, summarize the process of using partial fractions for integrating a rational function. Specify the algebraic tools needed throughout the process.
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