Set up the following integral using partial fractions after showing the long division, then integrate. A. SHOW LONG DIVISION STEPS; B. SHOW PARTIAL FRACTION DECOMPOSITION AND WORK; C. INTEGRATE AND SHOW WORK. - 6x³ +3x² – 143x -80 -dx x² - 25
Set up the following integral using partial fractions after showing the long division, then integrate. A. SHOW LONG DIVISION STEPS; B. SHOW PARTIAL FRACTION DECOMPOSITION AND WORK; C. INTEGRATE AND SHOW WORK. - 6x³ +3x² – 143x -80 -dx x² - 25
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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![## Partial Fractions and Integration
### Instructions
Set up the following integral using **partial fractions** after **showing the long division**, then integrate.
### Steps
**A. Show Long Division Steps**
**B. Show Partial Fraction Decomposition and Work**
**C. Integrate and Show Work**
\[
\int \frac{6x^3 + 3x^2 - 143x - 80}{x^2 - 25} \, dx
\]
### Explanation
- **Long Division**: Dividing the polynomial \(6x^3 + 3x^2 - 143x - 80\) by \(x^2 - 25\) to simplify the integral.
- **Partial Fraction Decomposition**: Breaking down the rational expression into simpler fractions that are easier to integrate.
- **Integration**: Calculating the integral of the decomposed fractions with respect to \(x\).
Each step should be detailed to ensure clarity in the process from division, through decomposition, to integration.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff4586ed3-251d-413f-881b-72e0763cbb19%2Ffbfb8916-b5bb-4cf0-913a-45aa281cf9a4%2Fbooybsi_processed.png&w=3840&q=75)
Transcribed Image Text:## Partial Fractions and Integration
### Instructions
Set up the following integral using **partial fractions** after **showing the long division**, then integrate.
### Steps
**A. Show Long Division Steps**
**B. Show Partial Fraction Decomposition and Work**
**C. Integrate and Show Work**
\[
\int \frac{6x^3 + 3x^2 - 143x - 80}{x^2 - 25} \, dx
\]
### Explanation
- **Long Division**: Dividing the polynomial \(6x^3 + 3x^2 - 143x - 80\) by \(x^2 - 25\) to simplify the integral.
- **Partial Fraction Decomposition**: Breaking down the rational expression into simpler fractions that are easier to integrate.
- **Integration**: Calculating the integral of the decomposed fractions with respect to \(x\).
Each step should be detailed to ensure clarity in the process from division, through decomposition, to integration.
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