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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![### Verification of Integration by Differentiation
To verify the formula, we differentiate the following integral:
\[
\int \frac{10}{10x - 11} \, dx = \ln |10x - 11| + C, \quad x \neq \frac{11}{10}
\]
The following steps outline this process:
1. **Differentiate the Result**: Apply the derivative to the right-hand side, \(\ln |10x - 11| + C\).
2. **Chain Rule Application**: Use the chain rule for differentiation on \(\ln |10x - 11|\).
3. **Confirm the Original Integrand**: The differentiation should yield the original integrand \(\frac{10}{10x - 11}\).
An interactive feature (dropdown menu) is provided for engaging with the verification process:
- **Dropdown Choice**: The \(\boxed{\text{differentiation}}\) dropdown can be used to verify the given formula.
This process ensures comprehension by connecting integration and differentiation concepts.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0adbc2a4-d159-4c5a-bcc6-33074c5d91f6%2Fe640b7cf-7e62-4180-a671-be663db0dc94%2Fwoqjky_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Verification of Integration by Differentiation
To verify the formula, we differentiate the following integral:
\[
\int \frac{10}{10x - 11} \, dx = \ln |10x - 11| + C, \quad x \neq \frac{11}{10}
\]
The following steps outline this process:
1. **Differentiate the Result**: Apply the derivative to the right-hand side, \(\ln |10x - 11| + C\).
2. **Chain Rule Application**: Use the chain rule for differentiation on \(\ln |10x - 11|\).
3. **Confirm the Original Integrand**: The differentiation should yield the original integrand \(\frac{10}{10x - 11}\).
An interactive feature (dropdown menu) is provided for engaging with the verification process:
- **Dropdown Choice**: The \(\boxed{\text{differentiation}}\) dropdown can be used to verify the given formula.
This process ensures comprehension by connecting integration and differentiation concepts.
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