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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![**Calculus Problem: Evaluating an Indefinite Integral**
In this problem, we are tasked with evaluating the indefinite integral given below:
\[ \int \frac{(\ln(x))^{20}}{x} \, dx \]
Here, \(\ln(x)\) represents the natural logarithm of \(x\), and we are interested in finding the indefinite integral of the function \(\frac{(\ln(x))^{20}}{x}\) with respect to \(x\).
### Step-by-Step Solution:
To solve this integral, we use the substitution method:
1. **Substitute \(u = \ln(x)\)**:
- Therefore, \(du = \frac{1}{x} \, dx\).
2. **Rewrite the integral in terms of \(u\)**:
\[ \int \frac{(\ln(x))^{20}}{x} \, dx = \int u^{20} \, du \]
3. **Evaluate the integral in terms of \(u\)**:
\[ \int u^{20} \, du = \frac{u^{21}}{21} + C \]
- Here, \(C\) represents the constant of integration.
4. **Substitute \(u = \ln(x)\) back into the result**:
\[ \frac{(\ln(x))^{21}}{21} + C \]
### Final Answer:
\[ \boxed{ \frac{(\ln(x))^{21}}{21} + C } \]
This represents the indefinite integral of the given function \(\frac{(\ln(x))^{20}}{x}\) with respect to \(x\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F38aaaeb0-3205-4fdb-bd84-df06c53da4aa%2Fff6b749c-7df6-4a96-9198-6d9196a3fd43%2Fyshstt_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Calculus Problem: Evaluating an Indefinite Integral**
In this problem, we are tasked with evaluating the indefinite integral given below:
\[ \int \frac{(\ln(x))^{20}}{x} \, dx \]
Here, \(\ln(x)\) represents the natural logarithm of \(x\), and we are interested in finding the indefinite integral of the function \(\frac{(\ln(x))^{20}}{x}\) with respect to \(x\).
### Step-by-Step Solution:
To solve this integral, we use the substitution method:
1. **Substitute \(u = \ln(x)\)**:
- Therefore, \(du = \frac{1}{x} \, dx\).
2. **Rewrite the integral in terms of \(u\)**:
\[ \int \frac{(\ln(x))^{20}}{x} \, dx = \int u^{20} \, du \]
3. **Evaluate the integral in terms of \(u\)**:
\[ \int u^{20} \, du = \frac{u^{21}}{21} + C \]
- Here, \(C\) represents the constant of integration.
4. **Substitute \(u = \ln(x)\) back into the result**:
\[ \frac{(\ln(x))^{21}}{21} + C \]
### Final Answer:
\[ \boxed{ \frac{(\ln(x))^{21}}{21} + C } \]
This represents the indefinite integral of the given function \(\frac{(\ln(x))^{20}}{x}\) with respect to \(x\).
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