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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Question
Am I doing it right?
Find the antiderivative of this equation.
![### Integration by Substitution Example
This image demonstrates a step-by-step process of integrating a rational function using the method of substitution.
#### Problem Statement
The integral to be solved is:
\[
-84500 \int \frac{1}{(x-3)^2} \, dx
\]
#### Solution Steps
1. **Substitution**
- Set \( u = x - 3 \), which implies \( du = dx \).
2. **Rewrite the Integral**
- The integral becomes:
\[
-84500 \int u^{-2} \, du
\]
3. **Integrate Using Power Rule**
- Apply the power rule for integration:
\[
\int u^{-2} \, du = -\frac{1}{u}
\]
4. **Include Constant Factor**
- The expression becomes:
\[
-84500 \left(-\frac{1}{u}\right)
\]
5. **Substitute Back**
- Replace \( u \) with \( x - 3 \):
\[
\frac{84500}{x-3}
\]
#### Conclusion
The final integrated result is:
\[
\frac{84500}{x-3}
\]
### Explanation
This method of integration by substitution is useful for simplifying integrals involving polynomials and rational functions, enabling more manageable expressions.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0e691e7d-3aa6-4d87-8ff6-0afc8b27fff3%2Ffa0f6282-4440-4c69-805f-3762653b52d7%2Fijn0gu_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Integration by Substitution Example
This image demonstrates a step-by-step process of integrating a rational function using the method of substitution.
#### Problem Statement
The integral to be solved is:
\[
-84500 \int \frac{1}{(x-3)^2} \, dx
\]
#### Solution Steps
1. **Substitution**
- Set \( u = x - 3 \), which implies \( du = dx \).
2. **Rewrite the Integral**
- The integral becomes:
\[
-84500 \int u^{-2} \, du
\]
3. **Integrate Using Power Rule**
- Apply the power rule for integration:
\[
\int u^{-2} \, du = -\frac{1}{u}
\]
4. **Include Constant Factor**
- The expression becomes:
\[
-84500 \left(-\frac{1}{u}\right)
\]
5. **Substitute Back**
- Replace \( u \) with \( x - 3 \):
\[
\frac{84500}{x-3}
\]
#### Conclusion
The final integrated result is:
\[
\frac{84500}{x-3}
\]
### Explanation
This method of integration by substitution is useful for simplifying integrals involving polynomials and rational functions, enabling more manageable expressions.
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