7.1 Integration By Parts/12

Evaluate the integral

${\begin{aligned}\int p^{5}ln(p)dp\\[2ex]\end{aligned}}$

${\begin{aligned}u=ln(p)\quad \quad dv=p^{5}dp\\[2ex]du={\frac {1}{p}}\quad \quad \quad v={\frac {1}{6}}p^{6}\\[2ex]\end{aligned}}$

${\begin{aligned}\int p^{5}ln(p)dp\quad =\quad {\frac {1}{6}}p^{6}ln(p)-\int p^{6}{\frac {1}{p}}dp\quad =\quad {\frac {1}{6}}p^{6}ln(p)-{\frac {1}{36}}p^{6}+C\end{aligned}}$