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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Question
![**Evaluate the following definite integral:**
\[ \int_{1}^{e} \left( 5x + \frac{4}{x} \right) dx \]
This integral involves the summation of two functions, \( 5x \) and \( \frac{4}{x} \), integrated with respect to \( x \) from 1 to \( e \). Here's a step-by-step explanation to solve this integral:
1. **Separate the integral into two parts:**
\[ \int_{1}^{e} \left( 5x + \frac{4}{x} \right) dx = \int_{1}^{e} 5x \, dx + \int_{1}^{e} \frac{4}{x} \, dx \]
2. **Evaluate each part separately:**
- For the first part \( \int_{1}^{e} 5x \, dx \):
Integrate \( 5x \) with respect to \( x \):
\[ \int 5x \, dx = \frac{5x^2}{2} \]
Evaluate this at the bounds \( 1 \) and \( e \):
\[ \left[ \frac{5x^2}{2} \right]_{1}^{e} = \frac{5e^2}{2} - \frac{5(1)^2}{2} = \frac{5e^2}{2} - \frac{5}{2} \]
- For the second part \( \int_{1}^{e} \frac{4}{x} \, dx \):
Integrate \( \frac{4}{x} \) with respect to \( x \):
\[ \int \frac{4}{x} \, dx = 4 \ln|x| \]
Evaluate this at the bounds \( 1 \) and \( e \):
\[ \left[ 4 \ln|x| \right]_{1}^{e} = 4 \ln(e) - 4 \ln(1) = 4 \cdot 1 - 4 \cdot 0 = 4 \]
3. **Combine the results from the two parts:**
\[ \frac{5e^2}{2} - \frac{5}{2} + 4 \]
Therefore,](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fbe634a2f-90fc-4c70-a49c-ac4a0564f81f%2Fe5201e0c-2615-4f55-abb2-4479977cd69c%2Fi3wzs58_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Evaluate the following definite integral:**
\[ \int_{1}^{e} \left( 5x + \frac{4}{x} \right) dx \]
This integral involves the summation of two functions, \( 5x \) and \( \frac{4}{x} \), integrated with respect to \( x \) from 1 to \( e \). Here's a step-by-step explanation to solve this integral:
1. **Separate the integral into two parts:**
\[ \int_{1}^{e} \left( 5x + \frac{4}{x} \right) dx = \int_{1}^{e} 5x \, dx + \int_{1}^{e} \frac{4}{x} \, dx \]
2. **Evaluate each part separately:**
- For the first part \( \int_{1}^{e} 5x \, dx \):
Integrate \( 5x \) with respect to \( x \):
\[ \int 5x \, dx = \frac{5x^2}{2} \]
Evaluate this at the bounds \( 1 \) and \( e \):
\[ \left[ \frac{5x^2}{2} \right]_{1}^{e} = \frac{5e^2}{2} - \frac{5(1)^2}{2} = \frac{5e^2}{2} - \frac{5}{2} \]
- For the second part \( \int_{1}^{e} \frac{4}{x} \, dx \):
Integrate \( \frac{4}{x} \) with respect to \( x \):
\[ \int \frac{4}{x} \, dx = 4 \ln|x| \]
Evaluate this at the bounds \( 1 \) and \( e \):
\[ \left[ 4 \ln|x| \right]_{1}^{e} = 4 \ln(e) - 4 \ln(1) = 4 \cdot 1 - 4 \cdot 0 = 4 \]
3. **Combine the results from the two parts:**
\[ \frac{5e^2}{2} - \frac{5}{2} + 4 \]
Therefore,
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