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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![### Calculus Exercise Problems
#### Problem 15
Evaluate the integral:
\[ \int \sin^3{x} \cos^4{x} \, dx \]
#### Problem 16
Evaluate the integral:
\[ \int \sec^4{x} \tan^4{x} \, dx \]
### Explanation:
You are asked to solve the above integration problems. For Problem 15, use trigonometric identities and possibly a substitution method to simplify the integral involving sine and cosine. Similarly, for Problem 16, utilize the properties of secant and tangent trigonometric functions for integration.
### Tips for Solving:
- For integrals involving powers of sine and cosine, a common technique is to use identities such as \( \sin^2{x} = 1 - \cos^2{x} \) to convert all terms to one trigonometric function type.
- For integrals involving secant and tangent, look for opportunities to use substitutions involving \( u = \tan{x} \) or \( u = \sec{x} \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffcbf1922-0087-4a8a-9a65-bc6f24870989%2Fcc6549c7-d75e-4392-98ba-346022f403fb%2F2xrv1ia.jpeg&w=3840&q=75)
Transcribed Image Text:### Calculus Exercise Problems
#### Problem 15
Evaluate the integral:
\[ \int \sin^3{x} \cos^4{x} \, dx \]
#### Problem 16
Evaluate the integral:
\[ \int \sec^4{x} \tan^4{x} \, dx \]
### Explanation:
You are asked to solve the above integration problems. For Problem 15, use trigonometric identities and possibly a substitution method to simplify the integral involving sine and cosine. Similarly, for Problem 16, utilize the properties of secant and tangent trigonometric functions for integration.
### Tips for Solving:
- For integrals involving powers of sine and cosine, a common technique is to use identities such as \( \sin^2{x} = 1 - \cos^2{x} \) to convert all terms to one trigonometric function type.
- For integrals involving secant and tangent, look for opportunities to use substitutions involving \( u = \tan{x} \) or \( u = \sec{x} \).
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