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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Question
![### Problem Statement
**Evaluate**
\[ \int \cos^3(7x) \sin(7x) \, dx \]
### Solution Approach
This problem involves finding the indefinite integral of a trigonometric function. The expression given is \(\cos^3(7x) \sin(7x)\).
One method to solve this problem is using a substitution technique. Let's consider the following steps:
1. **Substitution**:
Let \( u = \cos(7x) \). Then, \( \frac{du}{dx} = -7\sin(7x) \).
Hence, \( dx = \frac{du}{-7\sin(7x)} \).
2. **Rewriting the Integral**:
Substitute and rewrite the integral in terms of \( u \):
\[
\int \cos^3(7x) \sin(7x) \, dx = \int u^3 \left(-\frac{1}{7}\right) \, du
\]
3. **Integration**:
Integrate the expression with respect to \( u \):
\[
= -\frac{1}{7} \int u^3 \, du
= -\frac{1}{7} \cdot \frac{u^4}{4} + C
= -\frac{1}{28} u^4 + C
\]
4. **Back-substitution**:
Replace \( u \) with \( \cos(7x) \):
\[
= -\frac{1}{28} (\cos(7x))^4 + C
\]
### Final Answer
The integral evaluates to:
\[
-\frac{1}{28} \cos^4(7x) + C
\]
where \( C \) is the constant of integration.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F621ac9f8-2dbb-4715-ab7b-42c3ebc90b44%2Fe147950d-d3c8-4b2c-8cb2-a352d9f1fd0e%2Fm9bawfc_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Problem Statement
**Evaluate**
\[ \int \cos^3(7x) \sin(7x) \, dx \]
### Solution Approach
This problem involves finding the indefinite integral of a trigonometric function. The expression given is \(\cos^3(7x) \sin(7x)\).
One method to solve this problem is using a substitution technique. Let's consider the following steps:
1. **Substitution**:
Let \( u = \cos(7x) \). Then, \( \frac{du}{dx} = -7\sin(7x) \).
Hence, \( dx = \frac{du}{-7\sin(7x)} \).
2. **Rewriting the Integral**:
Substitute and rewrite the integral in terms of \( u \):
\[
\int \cos^3(7x) \sin(7x) \, dx = \int u^3 \left(-\frac{1}{7}\right) \, du
\]
3. **Integration**:
Integrate the expression with respect to \( u \):
\[
= -\frac{1}{7} \int u^3 \, du
= -\frac{1}{7} \cdot \frac{u^4}{4} + C
= -\frac{1}{28} u^4 + C
\]
4. **Back-substitution**:
Replace \( u \) with \( \cos(7x) \):
\[
= -\frac{1}{28} (\cos(7x))^4 + C
\]
### Final Answer
The integral evaluates to:
\[
-\frac{1}{28} \cos^4(7x) + C
\]
where \( C \) is the constant of integration.
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