Decompose the following fraction using a long division: 5x³ + 2x² - 38x - 21 x² - 9 a = b = 5 0 C = d = -21 9 A= M 1 B = 8 Use the partial fractions method on the resulting fraction to get: 5x³ + 2x² 38x - 21 A B ax+b+ where x² - 9 - S 500 = ax + b + Integrate term by term to find 5x³ 2x² 38x - 21 + x² - 9 - cx + d x² - 9 om X 3 where + x + 3 I 5 2 x² +8 ln(x-3) + ln x + 3 +C
Decompose the following fraction using a long division: 5x³ + 2x² - 38x - 21 x² - 9 a = b = 5 0 C = d = -21 9 A= M 1 B = 8 Use the partial fractions method on the resulting fraction to get: 5x³ + 2x² 38x - 21 A B ax+b+ where x² - 9 - S 500 = ax + b + Integrate term by term to find 5x³ 2x² 38x - 21 + x² - 9 - cx + d x² - 9 om X 3 where + x + 3 I 5 2 x² +8 ln(x-3) + ln x + 3 +C
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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![**Title: Decomposition and Integration of Fractions**
**Decompose the following fraction using a long division:**
\[ \frac{5x^3 + 2x^2 - 38x - 21}{x^2 - 9} \]
\[= ax + b + \frac{cx + d}{x^2 - 9}, \] where
\[ a = 5 \]
\[ b = 0 \]
\[ c = 9 \]
\[ d = -21 \]
**Use the partial fractions method on the resulting fraction to get:**
\[ \frac{5x^3 + 2x^2 - 38x - 21}{x^2 - 9} = ax + b + \frac{A}{x - 3} + \frac{B}{x + 3},\]
where
\[ A = 1 \]
\[ B = 8 \]
**Integrate term by term to find:**
\[ \int \frac{5x^3 + 2x^2 - 38x - 21}{x^2 - 9} \, dx \]
\[= \frac{5}{2} x^2 + 8 \ln|x - 3| + \ln|x + 3| + C \]
---
In essence, this content guides students through the process of decomposing and integrating a complex fraction by employing long division and partial fractions. The calculations for constants \(a\), \(b\), \(c\), and \(d\) are shown, followed by finding \(A\) and \(B\) for partial fractions, concluding with the integration.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe8fabc15-e5db-480c-94d9-e42b67b656a8%2Fe80de562-d20f-41ed-9bfb-2bf487760567%2F3b81y1_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Title: Decomposition and Integration of Fractions**
**Decompose the following fraction using a long division:**
\[ \frac{5x^3 + 2x^2 - 38x - 21}{x^2 - 9} \]
\[= ax + b + \frac{cx + d}{x^2 - 9}, \] where
\[ a = 5 \]
\[ b = 0 \]
\[ c = 9 \]
\[ d = -21 \]
**Use the partial fractions method on the resulting fraction to get:**
\[ \frac{5x^3 + 2x^2 - 38x - 21}{x^2 - 9} = ax + b + \frac{A}{x - 3} + \frac{B}{x + 3},\]
where
\[ A = 1 \]
\[ B = 8 \]
**Integrate term by term to find:**
\[ \int \frac{5x^3 + 2x^2 - 38x - 21}{x^2 - 9} \, dx \]
\[= \frac{5}{2} x^2 + 8 \ln|x - 3| + \ln|x + 3| + C \]
---
In essence, this content guides students through the process of decomposing and integrating a complex fraction by employing long division and partial fractions. The calculations for constants \(a\), \(b\), \(c\), and \(d\) are shown, followed by finding \(A\) and \(B\) for partial fractions, concluding with the integration.
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