1. Use the Average Value from a to b:
f avg = 1 b − a ∫ a b f ( x ) d x 1 12 ∫ 0 12 50 + 14 sin ( π 12 t ) d t = 1 12 [ 50 t − < m a t h > u = π 12 {\displaystyle f_{\text{avg}}={\frac {1}{b-a}}\int _{a}^{b}f(x)\,dx{\frac {1}{12}}\int _{0}^{12}50+14\sin({\frac {\pi }{12}}t)\,dt={\frac {1}{12}}[50t-<math>{\begin{aligned}u&={\frac {\pi }{12}}\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}}}
![{\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)