### Evaluate the Following Integral\[\int p^{6} \ln(p) \, dp\]This expression represents the integral of \( p^6 \ln(p) \) with respect to \( p \). The goal is to find the antiderivative or the function F(p) whose derivative is \( p^6 \ln(p) \). #### Steps to Solve:1. **Integration by Parts:** - Integration by parts formula: \(\int u \, dv = uv - \int v \, du\). - Choose \( u \) and \( dv \) appropriately based on the problem. - Let \( u = \ln(p) \), then \( du = \frac{1}{p} dp \). - Let \( dv = p^6 dp \), then \( v = \frac{p^7}{7} \).2. **Apply Integration by Parts:** \[ \int p^6 \ln(p) \, dp = \ln(p) \cdot \frac{p^7}{7} - \int \frac{p^7}{7} \cdot \frac{1}{p} \, dp \] Simplify the integral: \[ = \frac{p^7 \ln(p)}{7} - \int \frac{p^6}{7} \, dp \]3. **Evaluate the Remaining Integral:** \[ \int \frac{p^6}{7} \, dp = \frac{1}{7} \int p^6 \, dp = \frac{1}{7} \cdot \frac{p^7}{7} = \frac{p^7}{49} \]4. **Combine Results:** \[ \int p^6 \ln(p) \, dp = \frac{p^7 \ln(p)}{7} - \frac{p^7}{49} + C \] Simplify the expression: \[ = \frac{p^7 \ln(p)}{7} - \frac{p^7}{49} + C = \frac{p^7}{7} \left( \ln(p) - \frac{1}{7} \right) + C \]

Name: ### Evaluate the Following Integral\[\int p^{6} \ln(p) \, dp\]This ex
Uploaded: 2024-03-20T07:07:50.000Z
Channel: Bartleby
Description: ### Evaluate the Following Integral\[\int p^{6} \ln(p) \, dp\]This expression represents the integral of \( p^6 \ln(p) \) with respect to \( p \). The goal is to find the antiderivative or the function F(p) whose derivative is \( p^6 \ln(p) \). ####