Which expression can we use to solve Select the correct answer below: sec 0₁ sec² 0-1 tan 0 de of sec of se of se sec²0-sec²0 de sec² 0₁/sec² 0-1 sec 0 tan de sec² 0₁/sec² 0-1 de [x²√x²-10x² x² - 1 dx? Content attribution
Which expression can we use to solve Select the correct answer below: sec 0₁ sec² 0-1 tan 0 de of sec of se of se sec²0-sec²0 de sec² 0₁/sec² 0-1 sec 0 tan de sec² 0₁/sec² 0-1 de [x²√x²-10x² x² - 1 dx? Content attribution
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 Problem Explanation
#### Topic: Integration of Functions Involving Square Roots
---
**Problem Statement:**
Which expression can we use to solve the integral \( \int x^2 \sqrt{x^2 - 1} \, dx \) ?
**Select the correct answer below:**
1. \( \int \sec^4 \theta \sqrt{\sec^2 \theta - 1} \tan \theta \, d\theta \)
2. \( \int \sec^4 \theta - \sec^2 \theta \, d\theta \)
3. \( \int \sec^2 \theta \sqrt{\sec^2 \theta - 1} - \sec \theta \tan \theta \, d\theta \)
4. \( \int \sec^2 \theta \sqrt{\sec^2 \theta - 1} \, d\theta \)
---
This problem involves finding the appropriate substitution to transform the given integral into a more manageable form. The integral \( \int x^2 \sqrt{x^2 - 1} \, dx \) typically suggests a trigonometric substitution, such as \( x = \sec \theta \), to simplify the square root expression. Carefully analyze which of the given choices correctly represents this substitution and transformation.
**Answer Explanation:**
- When \( x = \sec \theta \), \( dx = \sec \theta \tan \theta \, d\theta \).
- The expression under the square root, \( \sqrt{x^2 - 1} \), transforms to \( \sqrt{\sec^2 \theta - 1} = \sqrt{\tan^2 \theta} = \tan \theta \).
Using the substitution:
\[ x = \sec \theta \]
\[ dx = \sec \theta \tan \theta \, d\theta \]
\[ \sqrt{x^2 - 1} = \tan \theta \]
The integral becomes:
\[ \int (\sec \theta)^2 (\tan \theta) (\sec \theta \tan \theta \, d\theta) \]
\[ \int \sec^4 \theta \tan^2 \theta \, d\theta \]
Hence, the correct transformed integral is aligned with one of the provided options.
Please select the correct option to proceed with solving the integral.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa79c18d0-6a0e-4a38-9496-04f82bd62d69%2Faab25874-7970-4cbd-839b-12892639517d%2Fw3dujx_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Calculus Problem Explanation
#### Topic: Integration of Functions Involving Square Roots
---
**Problem Statement:**
Which expression can we use to solve the integral \( \int x^2 \sqrt{x^2 - 1} \, dx \) ?
**Select the correct answer below:**
1. \( \int \sec^4 \theta \sqrt{\sec^2 \theta - 1} \tan \theta \, d\theta \)
2. \( \int \sec^4 \theta - \sec^2 \theta \, d\theta \)
3. \( \int \sec^2 \theta \sqrt{\sec^2 \theta - 1} - \sec \theta \tan \theta \, d\theta \)
4. \( \int \sec^2 \theta \sqrt{\sec^2 \theta - 1} \, d\theta \)
---
This problem involves finding the appropriate substitution to transform the given integral into a more manageable form. The integral \( \int x^2 \sqrt{x^2 - 1} \, dx \) typically suggests a trigonometric substitution, such as \( x = \sec \theta \), to simplify the square root expression. Carefully analyze which of the given choices correctly represents this substitution and transformation.
**Answer Explanation:**
- When \( x = \sec \theta \), \( dx = \sec \theta \tan \theta \, d\theta \).
- The expression under the square root, \( \sqrt{x^2 - 1} \), transforms to \( \sqrt{\sec^2 \theta - 1} = \sqrt{\tan^2 \theta} = \tan \theta \).
Using the substitution:
\[ x = \sec \theta \]
\[ dx = \sec \theta \tan \theta \, d\theta \]
\[ \sqrt{x^2 - 1} = \tan \theta \]
The integral becomes:
\[ \int (\sec \theta)^2 (\tan \theta) (\sec \theta \tan \theta \, d\theta) \]
\[ \int \sec^4 \theta \tan^2 \theta \, d\theta \]
Hence, the correct transformed integral is aligned with one of the provided options.
Please select the correct option to proceed with solving the integral.
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