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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![### Exercise: Evaluate an Improper Integral
Evaluate the improper integral:
\[ \int_{0}^{5} \frac{1}{(x - 1)^4} \, dx \]
#### Possible Answers:
1. \( \boxed{0} \)
2. \( \boxed{\frac{-1}{192} - \frac{1}{3}} \)
3. \( \boxed{\frac{-1}{192} + \frac{1}{3}} \)
4. \( \boxed{1} \)
5. \( \boxed{\text{The improper integral diverges.}} \)
### Explanation:
To evaluate the integral, consider any points within the interval \([0, 5]\) where the integrand may become undefined or unbounded. Here, the integrand \(\frac{1}{(x - 1)^4}\) becomes undefined at \(x = 1\). Thus, the integral is improper and requires special consideration.
For more details and solution steps, please refer to the section on Improper Integrals on our Educational website.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc88a4be5-ce0c-4e25-871d-a0bd16127b90%2Fa414b5ff-8c6f-4140-ad34-c6dcfae12deb%2Frdwy1y_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Exercise: Evaluate an Improper Integral
Evaluate the improper integral:
\[ \int_{0}^{5} \frac{1}{(x - 1)^4} \, dx \]
#### Possible Answers:
1. \( \boxed{0} \)
2. \( \boxed{\frac{-1}{192} - \frac{1}{3}} \)
3. \( \boxed{\frac{-1}{192} + \frac{1}{3}} \)
4. \( \boxed{1} \)
5. \( \boxed{\text{The improper integral diverges.}} \)
### Explanation:
To evaluate the integral, consider any points within the interval \([0, 5]\) where the integrand may become undefined or unbounded. Here, the integrand \(\frac{1}{(x - 1)^4}\) becomes undefined at \(x = 1\). Thus, the integral is improper and requires special consideration.
For more details and solution steps, please refer to the section on Improper Integrals on our Educational website.
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