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
![**Evaluate**
\[ \int \frac{2x + 1}{(4x + 3)(x + 1)} \, dx \]
---
In this integral, the function to be integrated is \( \frac{2x + 1}{(4x + 3)(x + 1)} \). The denominator is a product of two linear factors, \( 4x + 3 \) and \( x + 1 \). This is a rational function, which often requires partial fraction decomposition for integration.
**Partial Fraction Decomposition:**
The expression can be decomposed as:
\[ \frac{2x + 1}{(4x + 3)(x + 1)} = \frac{A}{4x + 3} + \frac{B}{x + 1} \]
To find constants \( A \) and \( B \), multiply through by the common denominator and solve the resulting system of equations.
**Integration:**
Once the partial fractions are determined, integrate each term separately:
\[ \int \frac{A}{4x + 3} \, dx + \int \frac{B}{x + 1} \, dx \]
**Result:**
The antiderivative can be found by integrating each term, typically resulting in a combination of logarithmic expressions.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3aff424b-0e96-4e90-b1d6-dc40303839e0%2Fe9606321-09ec-400a-9bfd-4e833116cb5f%2Fo62cxof_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Evaluate**
\[ \int \frac{2x + 1}{(4x + 3)(x + 1)} \, dx \]
---
In this integral, the function to be integrated is \( \frac{2x + 1}{(4x + 3)(x + 1)} \). The denominator is a product of two linear factors, \( 4x + 3 \) and \( x + 1 \). This is a rational function, which often requires partial fraction decomposition for integration.
**Partial Fraction Decomposition:**
The expression can be decomposed as:
\[ \frac{2x + 1}{(4x + 3)(x + 1)} = \frac{A}{4x + 3} + \frac{B}{x + 1} \]
To find constants \( A \) and \( B \), multiply through by the common denominator and solve the resulting system of equations.
**Integration:**
Once the partial fractions are determined, integrate each term separately:
\[ \int \frac{A}{4x + 3} \, dx + \int \frac{B}{x + 1} \, dx \]
**Result:**
The antiderivative can be found by integrating each term, typically resulting in a combination of logarithmic expressions.
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