Explanation

(c) $I_{av} = \frac{\int_{0}^{T/2}{i\ dt}}{\int_{0}^{T/2}{dt}} = \frac{\int_{0}^{T/2}{I_{0}\sin{(\omega t)dt}}}{T/2}$
$= \frac{2I_{0}}{T}\left\lbrack \frac{- \cos{\omega t}}{\omega} \right\rbrack_{0}^{T/2} = \frac{2I_{0}}{T}\left\lbrack - \frac{\cos\left( \frac{\omega T}{2} \right)}{\omega} + \frac{\cos{0{^\circ}}}{\omega} \right\rbrack$
$= \frac{2I_{0}}{\omega T}\left\lbrack - \cos\pi + \cos{0{^\circ}} \right\rbrack = \frac{2I_{0}}{2\pi}\lbrack 1 + 1\rbrack = \frac{2I_{0}}{\pi}$