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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![Title: Solving the Integral \( \int \frac{\cos (\ln x)}{x} \, dx \)
Introduction:
In this lesson, we will learn how to evaluate the following integral:
\[ \int \frac{\cos (\ln x)}{x} \, dx \]
Steps to Solve the Integral:
1. **Substitute\( \ln x = t\)**:
- First, we use the substitution \(t = \ln x\).
- Then, \( \frac{dt}{dx} = \frac{1}{x} \), and \( dx = x \, dt \).
2. **Rewrite the Integral**:
- Substitute \( t = \ln x \) and \( x \, dt = dx \) into the original integral:
\[ \int \frac{\cos (\ln x)}{x} \, dx = \int \cos(t) \, dt \]
3. **Integrate \( \cos(t) \)**:
- The integral of \( \cos(t) \) is:
\[ \int \cos(t) \, dt = \sin(t) + C \]
4. **Substitute Back \( t = \ln x \)**:
- Finally, we substitute back \( t = \ln x \) into the result:
\[ \sin(t) + C = \sin(\ln x) + C \]
Conclusion:
Therefore, the result of the integral \( \int \frac{\cos (\ln x)}{x} \, dx \) is:
\[ \sin(\ln x) + C \]
This concludes our lesson on how to evaluate the integral \( \int \frac{\cos (\ln x)}{x} \, dx \) by using substitution and basic integration techniques.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fddf14729-dbf2-4574-b793-bebd8df9c378%2Fd517fcf1-f8d4-4705-a444-32074ae38f6f%2Ffby409r_processed.png&w=3840&q=75)
Transcribed Image Text:Title: Solving the Integral \( \int \frac{\cos (\ln x)}{x} \, dx \)
Introduction:
In this lesson, we will learn how to evaluate the following integral:
\[ \int \frac{\cos (\ln x)}{x} \, dx \]
Steps to Solve the Integral:
1. **Substitute\( \ln x = t\)**:
- First, we use the substitution \(t = \ln x\).
- Then, \( \frac{dt}{dx} = \frac{1}{x} \), and \( dx = x \, dt \).
2. **Rewrite the Integral**:
- Substitute \( t = \ln x \) and \( x \, dt = dx \) into the original integral:
\[ \int \frac{\cos (\ln x)}{x} \, dx = \int \cos(t) \, dt \]
3. **Integrate \( \cos(t) \)**:
- The integral of \( \cos(t) \) is:
\[ \int \cos(t) \, dt = \sin(t) + C \]
4. **Substitute Back \( t = \ln x \)**:
- Finally, we substitute back \( t = \ln x \) into the result:
\[ \sin(t) + C = \sin(\ln x) + C \]
Conclusion:
Therefore, the result of the integral \( \int \frac{\cos (\ln x)}{x} \, dx \) is:
\[ \sin(\ln x) + C \]
This concludes our lesson on how to evaluate the integral \( \int \frac{\cos (\ln x)}{x} \, dx \) by using substitution and basic integration techniques.
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