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 4: Integration of Trigonometric Functions**
Evaluate the integral:
\[
\int \sin^3 x \cos^5 x \, dx
\]
**Explanation:**
This integral involves powers of sine and cosine, which typically require the application of trigonometric identities or substitution methods for simplification.
**Approach:**
1. **Use Trigonometric Identities:**
- Consider using identities like \(\sin^2 x = 1 - \cos^2 x\) or \(\cos^2 x = 1 - \sin^2 x\) to simplify the integrand.
2. **Substitution Method:**
- Choose a substitution that makes the integral easier to solve. For example, if you let \( u = \sin x \), then \( du = \cos x \, dx \).
**Note:**
This type of problem is common in calculus courses, particularly when learning about integration techniques involving trigonometric functions.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F648d38bc-f5b9-4baf-95bc-853621e34b39%2Faa71ce9a-09c3-4fa8-b03a-4dead6962f79%2F2br9ceo_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem 4: Integration of Trigonometric Functions**
Evaluate the integral:
\[
\int \sin^3 x \cos^5 x \, dx
\]
**Explanation:**
This integral involves powers of sine and cosine, which typically require the application of trigonometric identities or substitution methods for simplification.
**Approach:**
1. **Use Trigonometric Identities:**
- Consider using identities like \(\sin^2 x = 1 - \cos^2 x\) or \(\cos^2 x = 1 - \sin^2 x\) to simplify the integrand.
2. **Substitution Method:**
- Choose a substitution that makes the integral easier to solve. For example, if you let \( u = \sin x \), then \( du = \cos x \, dx \).
**Note:**
This type of problem is common in calculus courses, particularly when learning about integration techniques involving trigonometric functions.
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