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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![### Example Problem: Integrating Trigonometric Functions
Let's solve the following integral:
\[ \int 6 \sec^4 x \, dx \]
This problem involves integrating a trigonometric function raised to a power. To solve it, we can use a technique involving trigonometric identities or integration by substitution if applicable.
1. **Rewriting the Function:**
Express the integrand \( \sec^4 x \) in terms of basic trigonometric identities if needed. In this case, we can use the identity:
\[ \sec^2 x = 1 + \tan^2 x \]
2. **Applying Integration Techniques:**
Using the identity mentioned above or another suitable method:
\[ \int \sec^4 x \, dx = \int (\sec^2 x)^2 \, dx \]
You may continue with further substitution or use standard integration tables.
3. **Conducting the Integration:**
Apply the specific method to integrate the function. In some cases, an additional substitution like \( u = \tan x \) can simplify the problem, leading to:
\[ \int 6 (1 + \tan^2 x) \sec^2 x \, dx \]
4. **Returning to the Original Variable:**
After integrating, revert back to the variable \( x \) if any substitutions were used.
This problem showcases the importance of understanding trigonometric integration techniques, effectively using identities, and sometimes involving substitutions to simplify the integral.
**Note:** Detailed steps or computations may depend on the level of the course or the tools permitted (e.g., calculus techniques, computational tools).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F648d38bc-f5b9-4baf-95bc-853621e34b39%2F746e5e2a-bf3f-4926-9b06-c548ea1a6df3%2Fhl1uryx5_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Example Problem: Integrating Trigonometric Functions
Let's solve the following integral:
\[ \int 6 \sec^4 x \, dx \]
This problem involves integrating a trigonometric function raised to a power. To solve it, we can use a technique involving trigonometric identities or integration by substitution if applicable.
1. **Rewriting the Function:**
Express the integrand \( \sec^4 x \) in terms of basic trigonometric identities if needed. In this case, we can use the identity:
\[ \sec^2 x = 1 + \tan^2 x \]
2. **Applying Integration Techniques:**
Using the identity mentioned above or another suitable method:
\[ \int \sec^4 x \, dx = \int (\sec^2 x)^2 \, dx \]
You may continue with further substitution or use standard integration tables.
3. **Conducting the Integration:**
Apply the specific method to integrate the function. In some cases, an additional substitution like \( u = \tan x \) can simplify the problem, leading to:
\[ \int 6 (1 + \tan^2 x) \sec^2 x \, dx \]
4. **Returning to the Original Variable:**
After integrating, revert back to the variable \( x \) if any substitutions were used.
This problem showcases the importance of understanding trigonometric integration techniques, effectively using identities, and sometimes involving substitutions to simplify the integral.
**Note:** Detailed steps or computations may depend on the level of the course or the tools permitted (e.g., calculus techniques, computational tools).
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