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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![### Calculus Problem: Find the Integral
**Problem Statement:**
Evaluate the integral:
\[
\int \frac{-3x^2 + 9x + 21}{(x + 2)^2(2x + 1)} \, dx
\]
**Solution Layout:**
- Integrate the given rational function.
- Assume the solution includes a constant of integration, denoted as \(+ C\).
This exercise typically involves partial fraction decomposition to break down the integrand into simpler fractions that are easier to integrate.
**Steps to Solve:**
1. **Decompose into Partial Fractions:** Express the rational function as a sum of simpler fractions.
2. **Integrate Each Term:** Once decomposed, integrate each term separately.
3. **Combine Results:** Sum up the integrated terms and include the constant of integration \(+ C\).
Use algebraic manipulation and calculus rules for integration to find the complete solution.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8edd889e-338a-4275-872c-5b2b400b4e22%2Fae4ff0fc-ea82-4234-9106-c026615544b5%2Fpfvjhzq_processed.png&w=3840&q=75)
Transcribed Image Text:### Calculus Problem: Find the Integral
**Problem Statement:**
Evaluate the integral:
\[
\int \frac{-3x^2 + 9x + 21}{(x + 2)^2(2x + 1)} \, dx
\]
**Solution Layout:**
- Integrate the given rational function.
- Assume the solution includes a constant of integration, denoted as \(+ C\).
This exercise typically involves partial fraction decomposition to break down the integrand into simpler fractions that are easier to integrate.
**Steps to Solve:**
1. **Decompose into Partial Fractions:** Express the rational function as a sum of simpler fractions.
2. **Integrate Each Term:** Once decomposed, integrate each term separately.
3. **Combine Results:** Sum up the integrated terms and include the constant of integration \(+ C\).
Use algebraic manipulation and calculus rules for integration to find the complete solution.
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