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 the indefinite integral.
\[ \int 9 \sin^4(x) \cos(x) \,dx = \quad \boxed{} \quad + C \]
---
### Explanation
To solve this integral, we can use substitution method.
1. **Substitution:** Let \( u = \sin(x) \). Then, \( du = \cos(x) \,dx \).
2. **Transform the integral:**
\[
\int 9 \sin^4(x) \cos(x) \,dx \Rightarrow \int 9 u^4 \, du
\]
3. **Integrate:**
\[
\int 9 u^4 \, du = 9 \cdot \frac{u^5}{5} + C = \frac{9}{5} u^5 + C
\]
4. **Substitute \( u = \sin(x) \) back into the result:**
\[
\frac{9}{5} u^5 + C = \frac{9}{5} \sin^5(x) + C
\]
Thus, the evaluated indefinite integral is:
\[ \int 9 \sin^4(x) \cos(x) \,dx = \frac{9}{5} \sin^5(x) + C \]
---
This integral demonstrates the technique of using substitution to simplify the integration process. Here, recognizing the derivative relationship between \(\sin(x)\) and \(\cos(x)\) allows us to convert a trigonometric integral into a polynomial one, which is easier to solve.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5643f6a4-a7a4-4b03-93d4-ff1f1b43abe4%2Fab26a237-8b36-4ef5-8cdd-2c486f091d87%2Fdvlzfjm_processed.png&w=3840&q=75)
Transcribed Image Text:### Problem Statement
Evaluate the indefinite integral.
\[ \int 9 \sin^4(x) \cos(x) \,dx = \quad \boxed{} \quad + C \]
---
### Explanation
To solve this integral, we can use substitution method.
1. **Substitution:** Let \( u = \sin(x) \). Then, \( du = \cos(x) \,dx \).
2. **Transform the integral:**
\[
\int 9 \sin^4(x) \cos(x) \,dx \Rightarrow \int 9 u^4 \, du
\]
3. **Integrate:**
\[
\int 9 u^4 \, du = 9 \cdot \frac{u^5}{5} + C = \frac{9}{5} u^5 + C
\]
4. **Substitute \( u = \sin(x) \) back into the result:**
\[
\frac{9}{5} u^5 + C = \frac{9}{5} \sin^5(x) + C
\]
Thus, the evaluated indefinite integral is:
\[ \int 9 \sin^4(x) \cos(x) \,dx = \frac{9}{5} \sin^5(x) + C \]
---
This integral demonstrates the technique of using substitution to simplify the integration process. Here, recognizing the derivative relationship between \(\sin(x)\) and \(\cos(x)\) allows us to convert a trigonometric integral into a polynomial one, which is easier to solve.
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