1. Use the Average Value from a to b:
f avg = 1 b − a ∫ a b f ( x ) d x {\displaystyle f_{\text{avg}}={\frac {1}{b-a}}\int _{a}^{b}f(x)\,dx}
1 12 ∫ 0 12 50 + 14 sin ( π 12 t ) d t = 1 12 [ 50 t − 168 π cos ( π 12 t ) ] | 12 0 {\displaystyle {\frac {1}{12}}\int _{0}^{12}50+14\sin({\frac {\pi }{12}}t)\,dt={\frac {1}{12}}[50t-{\frac {168}{\pi }}\cos({\frac {\pi }{12}}t)]{\bigg |}_{12}^{0}}
u = π 12 t ∫ 14 sin ( u ) 12 π d u d t ⋅ d u d t = d t 14 ⋅ 12 π ∫ sin ( u ) d u 12 π d u = d t − 168 π cos ( u ) = − 168 π cos ( π 12 t ) {\displaystyle {\begin{aligned}u&={\frac {\pi }{12}}t\quad \quad \quad \quad \quad \int 14\sin(u){\frac {12}{\pi }}\,du\\\quad dt\cdot {\frac {du}{dt}}&=dt\quad \quad \quad \quad 14\cdot {\frac {12}{\pi }}\int \sin(u)\,du\\{\frac {12}{\pi }}du&=dt\quad \quad \quad \quad -{\frac {168}{\pi }}\cos(u)=-{\frac {168}{\pi }}\cos({\frac {\pi }{12}}t)\\\end{aligned}}}