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: Evaluating the Integral**
**Problem:**
Evaluate the integral.
\[ \int_{0}^{1} \left( x^{16} + 16^x \right) dx \]
**Provided Solution:**
\[ \frac{1}{12} + \frac{15}{\ln(16)} \]
**Explanation:**
This image depicts a mathematical problem where one is tasked with evaluating a definite integral. The integral in question is:
\[ \int_{0}^{1} \left( x^{16} + 16^x \right) dx \]
The provided solution is:
\[ \frac{1}{12} + \frac{15}{\ln(16)} \]
It is important to note that there is a red 'X' symbol next to the provided solution, indicating that the solution might be incorrect.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5ecf1484-f95f-4119-9858-43c8382b92c0%2F0f7ba31c-2227-4598-8051-8728b590ed76%2F9lg70i_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Title: Evaluating the Integral**
**Problem:**
Evaluate the integral.
\[ \int_{0}^{1} \left( x^{16} + 16^x \right) dx \]
**Provided Solution:**
\[ \frac{1}{12} + \frac{15}{\ln(16)} \]
**Explanation:**
This image depicts a mathematical problem where one is tasked with evaluating a definite integral. The integral in question is:
\[ \int_{0}^{1} \left( x^{16} + 16^x \right) dx \]
The provided solution is:
\[ \frac{1}{12} + \frac{15}{\ln(16)} \]
It is important to note that there is a red 'X' symbol next to the provided solution, indicating that the solution might be incorrect.
![### Problem Statement
Evaluate the integral:
\[ \int_{-7}^{7} \frac{4e^x}{\sinh(x) + \cosh(x)} \, dx \]
### Explanation of Terms and Symbols
- \(\int\) represents the integral symbol.
- The limits of integration are from \(-7\) to \(7\).
- The integrand (the function being integrated) is \(\frac{4e^x}{\sinh(x) + \cosh(x)}\).
- \(e\) is the base of the natural logarithm.
- \(x\) is the variable of integration.
- \(\sinh(x)\) and \(\cosh(x)\) are the hyperbolic sine and cosine functions, respectively.
To evaluate this integral, we need to understand the properties of the hyperbolic functions. Specifically, we use:
- \(\sinh(x) = \frac{e^x - e^{-x}}{2}\)
- \(\cosh(x) = \frac{e^x + e^{-x}}{2}\)
Simplifying the denominator:
\[ \sinh(x) + \cosh(x) = \frac{e^x - e^{-x}}{2} + \frac{e^x + e^{-x}}{2} = e^x \]
So the integral simplifies to:
\[ \int_{-7}^{7} \frac{4e^x}{e^x} \, dx = \int_{-7}^{7} 4 \, dx \]
Integrating \(4\) with respect to \(x\), we have:
\[ \int 4 \, dx = 4x \]
Evaluating this definite integral from \(-7\) to \(7\), we get:
\[ 4x \Big|_{-7}^{7} = 4(7) - 4(-7) = 28 + 28 = 56 \]
### Final Answer
The value of the integral is \(56\).
```markdown
**Explanation:**
- We first simplified the integrand by recognizing that \(\sinh(x) + \cosh(x) = e^x\).
- This significantly simplified the integral.
- Finally, upon evaluating the definite integral, we found the answer to be \(56\).
```](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5ecf1484-f95f-4119-9858-43c8382b92c0%2F0f7ba31c-2227-4598-8051-8728b590ed76%2Fbn6yrgp_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Problem Statement
Evaluate the integral:
\[ \int_{-7}^{7} \frac{4e^x}{\sinh(x) + \cosh(x)} \, dx \]
### Explanation of Terms and Symbols
- \(\int\) represents the integral symbol.
- The limits of integration are from \(-7\) to \(7\).
- The integrand (the function being integrated) is \(\frac{4e^x}{\sinh(x) + \cosh(x)}\).
- \(e\) is the base of the natural logarithm.
- \(x\) is the variable of integration.
- \(\sinh(x)\) and \(\cosh(x)\) are the hyperbolic sine and cosine functions, respectively.
To evaluate this integral, we need to understand the properties of the hyperbolic functions. Specifically, we use:
- \(\sinh(x) = \frac{e^x - e^{-x}}{2}\)
- \(\cosh(x) = \frac{e^x + e^{-x}}{2}\)
Simplifying the denominator:
\[ \sinh(x) + \cosh(x) = \frac{e^x - e^{-x}}{2} + \frac{e^x + e^{-x}}{2} = e^x \]
So the integral simplifies to:
\[ \int_{-7}^{7} \frac{4e^x}{e^x} \, dx = \int_{-7}^{7} 4 \, dx \]
Integrating \(4\) with respect to \(x\), we have:
\[ \int 4 \, dx = 4x \]
Evaluating this definite integral from \(-7\) to \(7\), we get:
\[ 4x \Big|_{-7}^{7} = 4(7) - 4(-7) = 28 + 28 = 56 \]
### Final Answer
The value of the integral is \(56\).
```markdown
**Explanation:**
- We first simplified the integrand by recognizing that \(\sinh(x) + \cosh(x) = e^x\).
- This significantly simplified the integral.
- Finally, upon evaluating the definite integral, we found the answer to be \(56\).
```
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