### Integral Calculation In this section, we will tackle the problem of evaluating the following integral: \[ \int \frac{x}{x^2 + x - 2} \, dx \] This integral appears to be a rational function where the numerator has a lower degree than the polynomial in the denominator. To approach this, we can start by factoring the denominator to simplify the integrand, possibly using partial fraction decomposition. **Steps to Solve the Integral:** 1. **Factor the denominator**: The polynomial \( x^2 + x - 2 \) can be factored into \((x+2)(x-1)\). \[ x^2 + x - 2 = (x+2)(x-1) \] 2. **Rewrite the integral**: Substitute the factored expression into the integrand: \[ \int \frac{x}{(x+2)(x-1)} \, dx \] 3. **Partial Fraction Decomposition**: We can use partial fractions to decompose \( \frac{x}{(x+2)(x-1)} \) into simpler fractions that are easier to integrate. \[ \frac{x}{(x+2)(x-1)} = \frac{A}{x+2} + \frac{B}{x-1} \] To find the constants \(A\) and \(B\), we will solve: \[ x = A(x-1) + B(x+2) \] By solving the system of equations, we determine the values of \(A\) and \(B\). 4. **Integrate each term**: Integrate the resulting partial fractions separately. 5. **Combine the results**: Sum the integrals of the partial fractions to find the final answer. Through these steps, students can systematically approach and solve the integral \( \int \frac{x}{x^2 + x - 2} \, dx \).
### Integral Calculation In this section, we will tackle the problem of evaluating the following integral: \[ \int \frac{x}{x^2 + x - 2} \, dx \] This integral appears to be a rational function where the numerator has a lower degree than the polynomial in the denominator. To approach this, we can start by factoring the denominator to simplify the integrand, possibly using partial fraction decomposition. **Steps to Solve the Integral:** 1. **Factor the denominator**: The polynomial \( x^2 + x - 2 \) can be factored into \((x+2)(x-1)\). \[ x^2 + x - 2 = (x+2)(x-1) \] 2. **Rewrite the integral**: Substitute the factored expression into the integrand: \[ \int \frac{x}{(x+2)(x-1)} \, dx \] 3. **Partial Fraction Decomposition**: We can use partial fractions to decompose \( \frac{x}{(x+2)(x-1)} \) into simpler fractions that are easier to integrate. \[ \frac{x}{(x+2)(x-1)} = \frac{A}{x+2} + \frac{B}{x-1} \] To find the constants \(A\) and \(B\), we will solve: \[ x = A(x-1) + B(x+2) \] By solving the system of equations, we determine the values of \(A\) and \(B\). 4. **Integrate each term**: Integrate the resulting partial fractions separately. 5. **Combine the results**: Sum the integrals of the partial fractions to find the final answer. Through these steps, students can systematically approach and solve the integral \( \int \frac{x}{x^2 + x - 2} \, dx \).
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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![### Integral Calculation
In this section, we will tackle the problem of evaluating the following integral:
\[ \int \frac{x}{x^2 + x - 2} \, dx \]
This integral appears to be a rational function where the numerator has a lower degree than the polynomial in the denominator. To approach this, we can start by factoring the denominator to simplify the integrand, possibly using partial fraction decomposition.
**Steps to Solve the Integral:**
1. **Factor the denominator**:
The polynomial \( x^2 + x - 2 \) can be factored into \((x+2)(x-1)\).
\[
x^2 + x - 2 = (x+2)(x-1)
\]
2. **Rewrite the integral**:
Substitute the factored expression into the integrand:
\[
\int \frac{x}{(x+2)(x-1)} \, dx
\]
3. **Partial Fraction Decomposition**:
We can use partial fractions to decompose \( \frac{x}{(x+2)(x-1)} \) into simpler fractions that are easier to integrate.
\[
\frac{x}{(x+2)(x-1)} = \frac{A}{x+2} + \frac{B}{x-1}
\]
To find the constants \(A\) and \(B\), we will solve:
\[
x = A(x-1) + B(x+2)
\]
By solving the system of equations, we determine the values of \(A\) and \(B\).
4. **Integrate each term**:
Integrate the resulting partial fractions separately.
5. **Combine the results**:
Sum the integrals of the partial fractions to find the final answer.
Through these steps, students can systematically approach and solve the integral \( \int \frac{x}{x^2 + x - 2} \, dx \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6c8732dd-627e-495c-a245-47d9f524edaa%2Fc81ee480-097a-44e0-af5a-caa35d78b345%2Fizc1lj_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Integral Calculation
In this section, we will tackle the problem of evaluating the following integral:
\[ \int \frac{x}{x^2 + x - 2} \, dx \]
This integral appears to be a rational function where the numerator has a lower degree than the polynomial in the denominator. To approach this, we can start by factoring the denominator to simplify the integrand, possibly using partial fraction decomposition.
**Steps to Solve the Integral:**
1. **Factor the denominator**:
The polynomial \( x^2 + x - 2 \) can be factored into \((x+2)(x-1)\).
\[
x^2 + x - 2 = (x+2)(x-1)
\]
2. **Rewrite the integral**:
Substitute the factored expression into the integrand:
\[
\int \frac{x}{(x+2)(x-1)} \, dx
\]
3. **Partial Fraction Decomposition**:
We can use partial fractions to decompose \( \frac{x}{(x+2)(x-1)} \) into simpler fractions that are easier to integrate.
\[
\frac{x}{(x+2)(x-1)} = \frac{A}{x+2} + \frac{B}{x-1}
\]
To find the constants \(A\) and \(B\), we will solve:
\[
x = A(x-1) + B(x+2)
\]
By solving the system of equations, we determine the values of \(A\) and \(B\).
4. **Integrate each term**:
Integrate the resulting partial fractions separately.
5. **Combine the results**:
Sum the integrals of the partial fractions to find the final answer.
Through these steps, students can systematically approach and solve the integral \( \int \frac{x}{x^2 + x - 2} \, dx \).
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