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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![### Integration Exercise
Evaluate the following definite integral:
\[
\int_{\ln\frac{\pi}{4}}^{\ln\frac{\pi}{2}} e^x \cos(e^x) \, dx
\]
In this problem, you are required to integrate the function \(e^x \cos(e^x)\) with respect to \(x\) over the interval from \(\ln\frac{\pi}{4}\) to \(\ln\frac{\pi}{2}\).
Understanding the notation:
- \(\ln\) denotes the natural logarithm.
- \(e\) represents Euler's number, the base of the natural logarithm.
- \(\cos\) is the cosine function, a trigonometric function.
#### Steps to Solve
1. **Identify the integral bounds:**
- The lower bound is \(\ln\frac{\pi}{4}\).
- The upper bound is \(\ln\frac{\pi}{2}\).
2. **Rewrite the function, if necessary, to simplify integration:**
- Consider using substitution if it helps to simplify the integral.
3. **Integrate the function:**
- Apply the appropriate integration techniques, such as substitution, by parts, or recognizing standard integrals.
4. **Evaluate the result at the bounds:**
- Substitute back the upper and lower limits of the integral and subtract the values.
### Example Solution
#### Substitution Method
Let \(u = e^x\), then \(du = e^x \, dx\) or equivalently \(dx = \frac{du}{u}\).
#### Change of Limits
When \(x = \ln\frac{\pi}{4}\), \(u = \frac{\pi}{4}\).
When \(x = \ln\frac{\pi}{2}\), \(u = \frac{\pi}{2}\).
So, the new integral bounds become:
\[ \int_{\frac{\pi}{4}}^{\frac{\pi}{2}} \cos(u) \, du \]
#### Integrate
The integral of \(\cos(u)\) is \(\sin(u)\):
\[ \left[ \sin(u) \right]_{\frac{\pi}{4}}^{\frac{\pi}{2}} \]
#### Evaluating at the Boundaries
\[ \sin\left(\frac{\pi}{2}\right) - \sin\left(\frac](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb4dca08d-d7c1-4511-81d7-e3b17171b463%2Fa6ab0c52-79bc-4fc3-b062-2ce79031afa6%2F1ubunvr_processed.png&w=3840&q=75)
Transcribed Image Text:### Integration Exercise
Evaluate the following definite integral:
\[
\int_{\ln\frac{\pi}{4}}^{\ln\frac{\pi}{2}} e^x \cos(e^x) \, dx
\]
In this problem, you are required to integrate the function \(e^x \cos(e^x)\) with respect to \(x\) over the interval from \(\ln\frac{\pi}{4}\) to \(\ln\frac{\pi}{2}\).
Understanding the notation:
- \(\ln\) denotes the natural logarithm.
- \(e\) represents Euler's number, the base of the natural logarithm.
- \(\cos\) is the cosine function, a trigonometric function.
#### Steps to Solve
1. **Identify the integral bounds:**
- The lower bound is \(\ln\frac{\pi}{4}\).
- The upper bound is \(\ln\frac{\pi}{2}\).
2. **Rewrite the function, if necessary, to simplify integration:**
- Consider using substitution if it helps to simplify the integral.
3. **Integrate the function:**
- Apply the appropriate integration techniques, such as substitution, by parts, or recognizing standard integrals.
4. **Evaluate the result at the bounds:**
- Substitute back the upper and lower limits of the integral and subtract the values.
### Example Solution
#### Substitution Method
Let \(u = e^x\), then \(du = e^x \, dx\) or equivalently \(dx = \frac{du}{u}\).
#### Change of Limits
When \(x = \ln\frac{\pi}{4}\), \(u = \frac{\pi}{4}\).
When \(x = \ln\frac{\pi}{2}\), \(u = \frac{\pi}{2}\).
So, the new integral bounds become:
\[ \int_{\frac{\pi}{4}}^{\frac{\pi}{2}} \cos(u) \, du \]
#### Integrate
The integral of \(\cos(u)\) is \(\sin(u)\):
\[ \left[ \sin(u) \right]_{\frac{\pi}{4}}^{\frac{\pi}{2}} \]
#### Evaluating at the Boundaries
\[ \sin\left(\frac{\pi}{2}\right) - \sin\left(\frac
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