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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![### Integral Evaluation Result
**Integral Expression**:
\[ \int \frac{x^2}{\sqrt{49 - x^2}} \, dx \]
**Solution**:
\[
\frac{49}{2} \arcsin\left(\frac{1}{7}x\right) - \frac{49}{2} \sin \left(2 \arcsin \left(\frac{1}{7}x \right) \right) + C
\]
**Explanation**:
The image shows an integral involving a rational function where the numerator is \( x^2 \) and the denominator is a square root \( \sqrt{49 - x^2} \).
The solution involves evaluating the integral and ends up expressing the result in terms of the arcsine (\(\arcsin\)) and sine (\(\sin\)) trigonometric functions. The constants in the solution appear due to integration operations and are combined into the final result.
The constant \( C \) represents the integration constant which signifies the family of antiderivatives of the given integrand. The presence of an extraneous red "X" at the bottom right of the solution might indicate an error or marking point from an educational tool or instructor but is irrelevant to the mathematical content.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5abad2ef-7d64-4544-b6f7-dd3dfbff74f8%2F26c6ef7d-b202-42b2-90f7-6015d1502e4b%2F8dh41t_processed.png&w=3840&q=75)
Transcribed Image Text:### Integral Evaluation Result
**Integral Expression**:
\[ \int \frac{x^2}{\sqrt{49 - x^2}} \, dx \]
**Solution**:
\[
\frac{49}{2} \arcsin\left(\frac{1}{7}x\right) - \frac{49}{2} \sin \left(2 \arcsin \left(\frac{1}{7}x \right) \right) + C
\]
**Explanation**:
The image shows an integral involving a rational function where the numerator is \( x^2 \) and the denominator is a square root \( \sqrt{49 - x^2} \).
The solution involves evaluating the integral and ends up expressing the result in terms of the arcsine (\(\arcsin\)) and sine (\(\sin\)) trigonometric functions. The constants in the solution appear due to integration operations and are combined into the final result.
The constant \( C \) represents the integration constant which signifies the family of antiderivatives of the given integrand. The presence of an extraneous red "X" at the bottom right of the solution might indicate an error or marking point from an educational tool or instructor but is irrelevant to the mathematical content.
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