Evaluate the integral.
∫ p 5 l n ( p ) d p {\displaystyle {\begin{aligned}\int p^{5}ln(p)dp\\[2ex]\end{aligned}}}
u = l n ( p ) d v = p 5 d p d u = 1 p v = 1 6 p 6 {\displaystyle {\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}}}
∫ p 5 l n ( p ) d p = 1 6 p 6 l n ( p ) − ∫ p 6 1 p d p = 1 6 p 6 l n ( p ) − 1 36 p 6 + C {\displaystyle {\begin{aligned}\int p^{5}ln(p)dp\\[2ex]&={\frac {1}{6}}p^{6}ln(p)-\int p^{6}{\frac {1}{p}}dp\\[2ex]&={\frac {1}{6}}p^{6}ln(p)-{\frac {1}{36}}p^{6}+C\end{aligned}}}
![{\displaystyle {\begin{aligned}\int p^{5}ln(p)dp\\[2ex]\end{aligned}}}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/a3912e6805b0b80a112b7a5d0d3550e4eb086010)
![{\displaystyle {\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}}}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/b68f7f9151aa266aea584d14de2e1310b53aa8db)
![{\displaystyle {\begin{aligned}\int p^{5}ln(p)dp\\[2ex]&={\frac {1}{6}}p^{6}ln(p)-\int p^{6}{\frac {1}{p}}dp\\[2ex]&={\frac {1}{6}}p^{6}ln(p)-{\frac {1}{36}}p^{6}+C\end{aligned}}}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/f4a0900b84456ce8b860232c47801bdb0925c2e5)