| (In(x))46 dx

Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
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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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### Evaluating the Indefinite Integral

**Problem Statement:**
Evaluate the indefinite integral. (Use \(C\) for the constant of integration.)

\[
\int \frac{(\ln(x))^{46}}{x} \, dx
\]

**Solution Approach:**

1. **Identify the integral form**: The given integral is \(\int \frac{(\ln(x))^{46}}{x} \, dx\).

2. **Substitution Method**:
   - Let \(u = \ln(x)\). Therefore, \(du = \frac{1}{x} \, dx\).

3. **Substitute into the integral**:
   - Replace \(\ln(x)\) with \(u\), and \(\frac{1}{x} \, dx\) with \(du\):
   \[
   \int u^{46} \, du
   \]

4. **Integrate with respect to \(u\)**:
   \[
   \int u^{46} \, du = \frac{u^{47}}{47} + C
   \]

5. **Substitute back \(u = \ln(x)\)**:
   \[
   \frac{(\ln(x))^{47}}{47} + C
   \]

Therefore, the evaluated indefinite integral is:
\[
\int \frac{(\ln(x))^{46}}{x} \, dx = \frac{(\ln(x))^{47}}{47} + C
\]

This step-by-step approach makes use of substitution techniques commonly taught in calculus courses.
Transcribed Image Text:### Evaluating the Indefinite Integral **Problem Statement:** Evaluate the indefinite integral. (Use \(C\) for the constant of integration.) \[ \int \frac{(\ln(x))^{46}}{x} \, dx \] **Solution Approach:** 1. **Identify the integral form**: The given integral is \(\int \frac{(\ln(x))^{46}}{x} \, dx\). 2. **Substitution Method**: - Let \(u = \ln(x)\). Therefore, \(du = \frac{1}{x} \, dx\). 3. **Substitute into the integral**: - Replace \(\ln(x)\) with \(u\), and \(\frac{1}{x} \, dx\) with \(du\): \[ \int u^{46} \, du \] 4. **Integrate with respect to \(u\)**: \[ \int u^{46} \, du = \frac{u^{47}}{47} + C \] 5. **Substitute back \(u = \ln(x)\)**: \[ \frac{(\ln(x))^{47}}{47} + C \] Therefore, the evaluated indefinite integral is: \[ \int \frac{(\ln(x))^{46}}{x} \, dx = \frac{(\ln(x))^{47}}{47} + C \] This step-by-step approach makes use of substitution techniques commonly taught in calculus courses.
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