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...
Related questions
Question
Evaluate using substitution
![### Problem Statement
Evaluate, using substitution:
\[
\int \frac{24t^5}{4t^6 + 3} \, dt.
\]
### Approach
To solve this integral using substitution, follow these steps:
1. **Identify the Inner Function**: Recognize that the denominator \(4t^6 + 3\) could be a good candidate for substitution.
2. **Substitution**: Let \( u = 4t^6 + 3 \). Then, find \( \frac{du}{dt} = 24t^5 \).
3. **Rewrite the Integral**: Substitute and rewrite the integral:
\[
\int \frac{24t^5}{4t^6 + 3} \, dt = \int \frac{1}{u} \, du
\]
4. **Integrate**: The integration of \(\frac{1}{u}\) is \(\ln |u| + C\).
5. **Back-Substitute**: Replace \( u \) with the original function:
\[
\ln |4t^6 + 3| + C
\]
### Conclusion
The evaluated integral is:
\[
\ln |4t^6 + 3| + C
\]
This method highlights the effective use of substitution in solving integrals, simplifying complex expressions into more manageable forms.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1b716114-3805-4246-a168-0232aeb2db3d%2Fb17a98fc-838f-4a5e-be4a-b0f42b79a068%2Ftvl7r6_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Problem Statement
Evaluate, using substitution:
\[
\int \frac{24t^5}{4t^6 + 3} \, dt.
\]
### Approach
To solve this integral using substitution, follow these steps:
1. **Identify the Inner Function**: Recognize that the denominator \(4t^6 + 3\) could be a good candidate for substitution.
2. **Substitution**: Let \( u = 4t^6 + 3 \). Then, find \( \frac{du}{dt} = 24t^5 \).
3. **Rewrite the Integral**: Substitute and rewrite the integral:
\[
\int \frac{24t^5}{4t^6 + 3} \, dt = \int \frac{1}{u} \, du
\]
4. **Integrate**: The integration of \(\frac{1}{u}\) is \(\ln |u| + C\).
5. **Back-Substitute**: Replace \( u \) with the original function:
\[
\ln |4t^6 + 3| + C
\]
### Conclusion
The evaluated integral is:
\[
\ln |4t^6 + 3| + C
\]
This method highlights the effective use of substitution in solving integrals, simplifying complex expressions into more manageable forms.
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