Which of the following are equivalent to T 1. II. III. A (В (С) D (3x+11) d'a (x+2)dx √₁² (³ 5 x + 2 [² (3 + 5 ) du U 3+ I only Il only Ill only II and III only dx 5 3x + 11 x + 2 -dx?
Which of the following are equivalent to T 1. II. III. A (В (С) D (3x+11) d'a (x+2)dx √₁² (³ 5 x + 2 [² (3 + 5 ) du U 3+ I only Il only Ill only II and III only dx 5 3x + 11 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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![**Topic: Evaluating Equivalent Integrals**
**Question:**
Which of the following are equivalent to
\[
\int_0^5 \frac{3x + 11}{x + 2} \, dx \, ?
\]
**Options:**
I. \(\int_0^5 (3x + 11) \, dx - \int_0^5 (x + 2) \, dx\)
II. \(\int_0^5 \left( 3 + \frac{5}{x + 2} \right) \, dx\)
III. \(\int_2^7 \left( 3 + \frac{5}{u} \right) \, du\)
**Choices:**
A) I only
B) II only
C) III only
D) II and III only
---
**Solution Explanation:**
To find equivalency, we transform the original integral. The given expression \(\int_0^5 \frac{3x + 11}{x + 2} \, dx\) can be broken down and compared with the options:
- **Option I** states that the given integral can be split into two separate integrals. This is incorrect because splitting the integrand in such a manner does not preserve the original structure of the rational function.
- **Option II** correctly simplifies the integrand using polynomial division: \(3x+11 = (3)(x + 2) + 5\). Thus, \(\frac{3x + 11}{x + 2} = 3 + \frac{5}{x + 2}\). This matches Option II.
- **Option III** involves changing the variable of integration from \(x\) to \(u\), with limits adjusted appropriately if \(u = x + 2\), making \(u\) range from 2 to 7. This means Option III is representing the same integral as Option II under a substitution.
Thus, both Option II and Option III are equivalent to the original integral.
**Correct Answer:** D) II and III only](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F019fb4ae-aebc-4c9c-a0ae-00e747a06944%2F119daea0-469e-4bf5-92ad-5ff41b8bed29%2Feailo8_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Topic: Evaluating Equivalent Integrals**
**Question:**
Which of the following are equivalent to
\[
\int_0^5 \frac{3x + 11}{x + 2} \, dx \, ?
\]
**Options:**
I. \(\int_0^5 (3x + 11) \, dx - \int_0^5 (x + 2) \, dx\)
II. \(\int_0^5 \left( 3 + \frac{5}{x + 2} \right) \, dx\)
III. \(\int_2^7 \left( 3 + \frac{5}{u} \right) \, du\)
**Choices:**
A) I only
B) II only
C) III only
D) II and III only
---
**Solution Explanation:**
To find equivalency, we transform the original integral. The given expression \(\int_0^5 \frac{3x + 11}{x + 2} \, dx\) can be broken down and compared with the options:
- **Option I** states that the given integral can be split into two separate integrals. This is incorrect because splitting the integrand in such a manner does not preserve the original structure of the rational function.
- **Option II** correctly simplifies the integrand using polynomial division: \(3x+11 = (3)(x + 2) + 5\). Thus, \(\frac{3x + 11}{x + 2} = 3 + \frac{5}{x + 2}\). This matches Option II.
- **Option III** involves changing the variable of integration from \(x\) to \(u\), with limits adjusted appropriately if \(u = x + 2\), making \(u\) range from 2 to 7. This means Option III is representing the same integral as Option II under a substitution.
Thus, both Option II and Option III are equivalent to the original integral.
**Correct Answer:** D) II and III only
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