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 Problem
#### Problem Statement:
(5) \(\int_{2}^{e} \frac{1}{x \sqrt{\ln{x}}} \, dx\)
In this problem, you are asked to evaluate the definite integral of the function \(\frac{1}{x \sqrt{\ln{x}}}\) with respect to \(x\), from the lower limit of 2 to the upper limit of \(e\), where \(e\) is the base of the natural logarithm, approximately equal to 2.71828.
### Graphs or Diagrams:
**There are no graphs or diagrams accompanying this problem. The focus is solely on evaluating the given integral.**
### Detailed Explanation:
This integral might require substitution or recognizing a pattern related to known antiderivatives. The presence of \( \ln{x} \) inside the square root suggests that a substitution involving \( u = \ln{x} \) might be helpful.
#### Potential Steps for Solution:
1. **Substitution**: Let \( u = \ln{x} \). This implies \( du = \frac{1}{x} dx \).
2. **Transform Limits**: When \( x = 2 \), \( \ln{2} = u_1 \). When \( x = e \), \( \ln{e} = 1 = u_2 \).
3. **Integral in terms of \( u \)**: Rewrite the integral using \( u \):
\[
\int_{2}^{e} \frac{1}{x \sqrt{\ln{x}}} \, dx = \int_{\ln{2}}^{1} \frac{1}{\sqrt{u}} \, du
\]
4. **Simplify and Integrate**: The integral now becomes:
\[
\int_{\ln{2}}^{1} u^{-\frac{1}{2}} \, du
\]
5. **Evaluate the Antiderivative**:
\[
\int u^{-\frac{1}{2}} \, du = 2u^{\frac{1}{2}} + C
\]
6. **Apply the Limits**:
\[
\left[ 2u^{\frac{1}{2}} \right]_{\ln{2}}^{1} = 2 \cdot (1^{\frac{1}{](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F715a84de-dd1a-49bc-bb04-088748385741%2F958a3dc9-7cb0-4781-9943-493e26aaabe7%2F3kk062_processed.png&w=3840&q=75)
Transcribed Image Text:### Integral Problem
#### Problem Statement:
(5) \(\int_{2}^{e} \frac{1}{x \sqrt{\ln{x}}} \, dx\)
In this problem, you are asked to evaluate the definite integral of the function \(\frac{1}{x \sqrt{\ln{x}}}\) with respect to \(x\), from the lower limit of 2 to the upper limit of \(e\), where \(e\) is the base of the natural logarithm, approximately equal to 2.71828.
### Graphs or Diagrams:
**There are no graphs or diagrams accompanying this problem. The focus is solely on evaluating the given integral.**
### Detailed Explanation:
This integral might require substitution or recognizing a pattern related to known antiderivatives. The presence of \( \ln{x} \) inside the square root suggests that a substitution involving \( u = \ln{x} \) might be helpful.
#### Potential Steps for Solution:
1. **Substitution**: Let \( u = \ln{x} \). This implies \( du = \frac{1}{x} dx \).
2. **Transform Limits**: When \( x = 2 \), \( \ln{2} = u_1 \). When \( x = e \), \( \ln{e} = 1 = u_2 \).
3. **Integral in terms of \( u \)**: Rewrite the integral using \( u \):
\[
\int_{2}^{e} \frac{1}{x \sqrt{\ln{x}}} \, dx = \int_{\ln{2}}^{1} \frac{1}{\sqrt{u}} \, du
\]
4. **Simplify and Integrate**: The integral now becomes:
\[
\int_{\ln{2}}^{1} u^{-\frac{1}{2}} \, du
\]
5. **Evaluate the Antiderivative**:
\[
\int u^{-\frac{1}{2}} \, du = 2u^{\frac{1}{2}} + C
\]
6. **Apply the Limits**:
\[
\left[ 2u^{\frac{1}{2}} \right]_{\ln{2}}^{1} = 2 \cdot (1^{\frac{1}{
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