Answer
$T_{avg}=(50+\frac{28}{\pi})^{\circ} F\approx 59^{\circ} F$
Work Step by Step
$T_{avg}=\frac{1}{12}\int_{0}^{12}(50+14sin(\frac{\pi t}{12}))dt$
$T_{avg}=\frac{1}{12}(\int_{0}^{12}(50)dt + \int_{0}^{12}(14sin(\frac{\pi t}{12}))dt)$
$T_{avg}=\frac{1}{12}((50x)\Big|_{0}^{12} + 14\int_{0}^{12}(sin(\frac{\pi t}{12}))dt)$
Solve the Indefinite Integral
$\int(sin(\frac{\pi t}{12}))dt$
Let u equal to $\frac{\pi t}{12}$
$du = \frac{\pi}{12}dt$
$\frac{\pi}{12}du = dt$
$\frac{12}{\pi}\int(sin({u}))du$
$-\frac{12}{\pi}(cos({u}))$
Substitute u and place it back into the $T_{avg}$ equation
$T_{avg}=\frac{1}{12}((50x)\Big|_{0}^{12} + 14 (\frac{-12}{\pi})(cos(\frac{\pi t}{12}))\Big|_{0}^{12})$
$T_{avg}=\frac{1}{12}((50x)\Big|_{0}^{12} - (\frac{168}{\pi})(cos(\frac{\pi t}{12}))\Big|_{0}^{12})$
$T_{avg}=\frac{1}{12}((50x) - (\frac{168}{\pi})(cos(\frac{\pi t}{12}))\Big|_{0}^{12})$
$T_{avg}=\frac{1}{12}(50(12)-\frac{168cos(\pi)}{\pi})-(50(0)-\frac{168cos(0)}{\pi})$
$T_{avg}=\frac{1}{12}(600)-\frac{168(-1)}{\pi})-(0-\frac{168(1)}{\pi})$
$T_{avg}=\frac{1}{12}(600)+\frac{336}{\pi})$
$T_{avg}=(50+\frac{28}{\pi})^{\circ} F\approx 59^{\circ} F$
