12 y 3= .2 +5x+6 4- 2. 1. 2 4 3.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![**Using Partial Fractions to Find the Area of the Given Region**
The given mathematical function is:
\[ y = \frac{12}{x^2 + 5x + 6} \]
This function is depicted in the accompanying graph, which shows its behavior over a specific range of \(x\) values.
### Graph Explanation:
- **Axes**: The graph is plotted with **y-axis** representing the dependent variable \(y\) and the **x-axis** representing the independent variable \(x\).
- **Range**: The \(x\) values range from 1 to 5 on the graph, with increments marked on the axis.
- **Function Behavior**:
- At \(x = 1\), the \(y\) value is around 4.
- As \(x\) increases, the \(y\) value decreases.
- By the time \(x = 5\), the \(y\) value is approaching 0 but never actually reaches it (asymptotic behavior).
- **Curve**: The curve starts from a higher value at \(y\), slopes downward steeply at first and then gradually flattens as \(x\) increases.
### Steps to Find the Area Using Partial Fractions:
To find the area under the curve of \(y = \frac{12}{x^2 + 5x + 6}\), you would:
1. **Factorize the Denominator**:
The quadratic \( x^2 + 5x + 6 \) can be factored into \( (x + 2)(x + 3) \).
2. **Express as Partial Fractions**:
Decompose \( \frac{12}{(x + 2)(x + 3)} \) into simpler fractions:
\[
\frac{12}{(x + 2)(x + 3)} = \frac{A}{x + 2} + \frac{B}{x + 3}
\]
Solve for constants \( A \) and \( B \).
3. **Integrate the Function**:
Integrate each term separately within your bounds of interest (i.e., from \(x = 1\) to \(x = 5\)) to find the area under the curve.
\[
\int_1^5 \left( \frac{](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fad118f04-c8c0-46fe-8652-72d195094665%2Fe50fe67c-c444-4314-8bf3-beb239668dbc%2Fbvt0xke_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Using Partial Fractions to Find the Area of the Given Region**
The given mathematical function is:
\[ y = \frac{12}{x^2 + 5x + 6} \]
This function is depicted in the accompanying graph, which shows its behavior over a specific range of \(x\) values.
### Graph Explanation:
- **Axes**: The graph is plotted with **y-axis** representing the dependent variable \(y\) and the **x-axis** representing the independent variable \(x\).
- **Range**: The \(x\) values range from 1 to 5 on the graph, with increments marked on the axis.
- **Function Behavior**:
- At \(x = 1\), the \(y\) value is around 4.
- As \(x\) increases, the \(y\) value decreases.
- By the time \(x = 5\), the \(y\) value is approaching 0 but never actually reaches it (asymptotic behavior).
- **Curve**: The curve starts from a higher value at \(y\), slopes downward steeply at first and then gradually flattens as \(x\) increases.
### Steps to Find the Area Using Partial Fractions:
To find the area under the curve of \(y = \frac{12}{x^2 + 5x + 6}\), you would:
1. **Factorize the Denominator**:
The quadratic \( x^2 + 5x + 6 \) can be factored into \( (x + 2)(x + 3) \).
2. **Express as Partial Fractions**:
Decompose \( \frac{12}{(x + 2)(x + 3)} \) into simpler fractions:
\[
\frac{12}{(x + 2)(x + 3)} = \frac{A}{x + 2} + \frac{B}{x + 3}
\]
Solve for constants \( A \) and \( B \).
3. **Integrate the Function**:
Integrate each term separately within your bounds of interest (i.e., from \(x = 1\) to \(x = 5\)) to find the area under the curve.
\[
\int_1^5 \left( \frac{
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