an anti derivative of 15 F(X) = (x²) F(x) = (x²) Explain why or why not
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
![**Problem Statement:**
Is \( F(x) = e^{(x^2)} \) an antiderivative of \( f(x) = e^{(x^2)} \)?
**Task:**
Explain why or why not.
**Explanation:**
To determine if \( F(x) = e^{x^2} \) is an antiderivative of \( f(x) = e^{x^2} \), we need to check if the derivative of \( F(x) \) results in \( f(x) \).
1. **Derivative of \( F(x) = e^{x^2} \):**
To differentiate \( F(x) \), use the chain rule:
\[
\frac{d}{dx} e^{x^2} = e^{x^2} \cdot \frac{d}{dx}(x^2) = e^{x^2} \cdot 2x = 2xe^{x^2}
\]
2. **Comparison with \( f(x) \):**
The derivative, \( 2xe^{x^2} \), is not equal to \( f(x) = e^{x^2} \).
**Conclusion:**
Therefore, \( F(x) = e^{x^2} \) is not an antiderivative of \( f(x) = e^{x^2} \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd6b4bd3b-c23c-4e19-9ff7-b724dd0e50b1%2F44ce8ec4-3b83-46ac-a902-53ea6b582d2e%2Fkxh0xsb_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Is \( F(x) = e^{(x^2)} \) an antiderivative of \( f(x) = e^{(x^2)} \)?
**Task:**
Explain why or why not.
**Explanation:**
To determine if \( F(x) = e^{x^2} \) is an antiderivative of \( f(x) = e^{x^2} \), we need to check if the derivative of \( F(x) \) results in \( f(x) \).
1. **Derivative of \( F(x) = e^{x^2} \):**
To differentiate \( F(x) \), use the chain rule:
\[
\frac{d}{dx} e^{x^2} = e^{x^2} \cdot \frac{d}{dx}(x^2) = e^{x^2} \cdot 2x = 2xe^{x^2}
\]
2. **Comparison with \( f(x) \):**
The derivative, \( 2xe^{x^2} \), is not equal to \( f(x) = e^{x^2} \).
**Conclusion:**
Therefore, \( F(x) = e^{x^2} \) is not an antiderivative of \( f(x) = e^{x^2} \).
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