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Evaluate the integral. (x16 + 16*)dx 1 15 + 12 In(16)

Evaluate the integral. (x16 + 16*)dx 1 15 + 12 In(16)

Calculus: Early Transcendentals
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.
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 \(56\). ```
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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