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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![**Problem Statement:**
Find a function with derivative equal to \( g(x) = \cos(x) e^{5\sin(x)} \).
**Explanation:**
To solve this problem, you need to integrate the given function \( g(x) \). The expression \( g(x) = \cos(x) e^{5\sin(x)} \) involves a combination of trigonometric and exponential functions.
**Steps for Solution:**
1. **Recognize the Function Type:**
- Notice that the function \( g(x) \) is in the form of a product of a cosine function and an exponential function dependent on sine.
2. **Integration Technique:**
- To find the antiderivative, consider using substitution methods due to the composition of functions.
3. **Substitution:**
- Let \( u = 5\sin(x) \), then \( \frac{du}{dx} = 5\cos(x) \). Thus, \( du = 5\cos(x) \, dx \) or \( \cos(x) \, dx = \frac{1}{5}du \).
4. **Integrate:**
- Rewrite the integral:
\[ \int \cos(x) e^{5\sin(x)} \, dx = \int e^{u} \cdot \frac{1}{5} \, du \]
\[ = \frac{1}{5} \int e^{u} \, du \]
\[ = \frac{1}{5} e^{u} + C \]
- Substitute back \( u = 5\sin(x) \):
\[ = \frac{1}{5} e^{5\sin(x)} + C \]
5. **Conclusion:**
- The function whose derivative is equal to \( g(x) = \cos(x) e^{5\sin(x)} \) is:
\[ F(x) = \frac{1}{5} e^{5\sin(x)} + C \]
- Where \( C \) is the constant of integration.
This approach demonstrates how to handle the integration of complex functions using substitution, a valuable technique in calculus to simplify the antiderivative process.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F23e6da21-480e-4db6-b0c9-f8fc2a559f65%2F0d67042e-d9d4-4f0d-9ab5-b3d43dd13832%2Fucn4r6a_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Find a function with derivative equal to \( g(x) = \cos(x) e^{5\sin(x)} \).
**Explanation:**
To solve this problem, you need to integrate the given function \( g(x) \). The expression \( g(x) = \cos(x) e^{5\sin(x)} \) involves a combination of trigonometric and exponential functions.
**Steps for Solution:**
1. **Recognize the Function Type:**
- Notice that the function \( g(x) \) is in the form of a product of a cosine function and an exponential function dependent on sine.
2. **Integration Technique:**
- To find the antiderivative, consider using substitution methods due to the composition of functions.
3. **Substitution:**
- Let \( u = 5\sin(x) \), then \( \frac{du}{dx} = 5\cos(x) \). Thus, \( du = 5\cos(x) \, dx \) or \( \cos(x) \, dx = \frac{1}{5}du \).
4. **Integrate:**
- Rewrite the integral:
\[ \int \cos(x) e^{5\sin(x)} \, dx = \int e^{u} \cdot \frac{1}{5} \, du \]
\[ = \frac{1}{5} \int e^{u} \, du \]
\[ = \frac{1}{5} e^{u} + C \]
- Substitute back \( u = 5\sin(x) \):
\[ = \frac{1}{5} e^{5\sin(x)} + C \]
5. **Conclusion:**
- The function whose derivative is equal to \( g(x) = \cos(x) e^{5\sin(x)} \) is:
\[ F(x) = \frac{1}{5} e^{5\sin(x)} + C \]
- Where \( C \) is the constant of integration.
This approach demonstrates how to handle the integration of complex functions using substitution, a valuable technique in calculus to simplify the antiderivative process.
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