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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![### Problem Statement:
**Integrate, rounding to three decimal places as needed:**
\[
\int_8^{12} \frac{12}{x^2 - 36} \, dx
\]
### Explanation:
To solve this integral, follow these steps:
1. **Identify the form**: Recognize that the integrand has a factor in the denominator that can be rewritten as a difference of squares.
2. **Partial Fraction Decomposition**: If necessary, decompose the fraction into simpler terms that can be integrated individually.
3. **Integration**: Integrate each part separately, applying appropriate integration techniques such as substitution.
4. **Compute the definite integral**: Evaluate the antiderivative at the upper and lower limits of integration.
5. **Round the result**: Round the final answer to three decimal places.
By following these steps, you should be able to compute the integral and provide an accurate, rounded answer suitable for educational purposes.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb7c72b9f-0930-4803-9715-bedcb9f3be97%2F6441fd12-6b6c-4ea4-9a5b-51265254a250%2Fxmtyy24_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Problem Statement:
**Integrate, rounding to three decimal places as needed:**
\[
\int_8^{12} \frac{12}{x^2 - 36} \, dx
\]
### Explanation:
To solve this integral, follow these steps:
1. **Identify the form**: Recognize that the integrand has a factor in the denominator that can be rewritten as a difference of squares.
2. **Partial Fraction Decomposition**: If necessary, decompose the fraction into simpler terms that can be integrated individually.
3. **Integration**: Integrate each part separately, applying appropriate integration techniques such as substitution.
4. **Compute the definite integral**: Evaluate the antiderivative at the upper and lower limits of integration.
5. **Round the result**: Round the final answer to three decimal places.
By following these steps, you should be able to compute the integral and provide an accurate, rounded answer suitable for educational purposes.
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