Single Variable Calculus, Chapter 8, 8.1, Section 8.1, Problem 12

Evaluate $\displaystyle \int p^5 \ln p dp$
If we let $u = \ln p$ and $dv = p^5 d_p$, then
$\displaystyle du = \frac{d_p}{p} \text{ and } v = \int p^5 d_p = \frac{p^6}{6}$

So,

$
\begin{equation}
\begin{aligned}
\int p^5 \ln p d_p &= uv - \int v du = \frac{p^6}{6} \ln p - \int \frac{p^6}{6} \left( \frac{d_p}{} \right)\\
\\
&= \frac{p^6}{6} \ln p - \frac{1}{6} \int p^6 d_p\\
\\
&= \frac{p^6}{6} \ln p - \frac{1}{6} \left[ \frac{p^6}{6} \right] + c\\
\\
&= \frac{p^6}{6} \left[ \ln p - \frac{1}{6} \right] + c
\end{aligned}
\end{equation}
$