c
(c)\({I_{av}} = \frac{{\int_0^{T/2} {\,\,\,\,\,i\,dt} }}{{\int_0^{T/2} {\,\,\,\,\,dt} }}\)\( = \frac{{\int_0^{T/2} {\,\,\,\,\,{I_0}\sin (\omega \,t)dt} }}{{T/2}}\)
\( = \frac{{2{I_0}}}{T}\left[ {\frac{{ - \cos \omega \,t}}{\omega }} \right]_0^{T/2}\)\( = \frac{{2{I_0}}}{T}\left[ { - \frac{{\cos \,\left( {\frac{{\omega T}}{2}} \right)}}{\omega } + \frac{{\cos {0^o}}}{\omega }} \right]\)
\( = \frac{{2{I_0}}}{{\omega \,T}}[ - \cos \pi + \cos {0^o}]\)\( = \frac{{2{I_0}}}{{2\pi }}[1 + 1] = \frac{{2{I_0}}}{\pi }\)