#f'(x)=p*sin(x/2), (p" const.)"#
By the Definition of Integral, #f(x)=intf'(x)dx+c#,
#=intp*sin(x/2)dx+c#,
#=p*intsin(x/2)dx+c#,
#=p{-cos(x/2)/(1/2)}+c#.
#rArr f(x)=-2pcos(x/2)+c.............(ast)#.
Given that, #f(0)=1, (ast) rArr 1=-2pcos0+c#.
#:. c-2p=1..................................................................(ast_1)#.
Next, #f(2pi)=0, (ast) rArr 0=-2pcos(2pi/2)+c#.
#:. c+2p=0.................................................................(ast_2)#.
Solving #(ast_1) and (ast_2), c=1/2, p=-1/4#.
# rArr f(x)=1/2{cos(x/2)+1}#.
