integral sqrt(1-cos(ax)) dx

cos(2ax)=1-2sin²(ax)

cos(ax)=1-2sin²(ax/2)

1-cos(ax)=2sin²(ax/2)

sqrt(1-cos(x))=+[sqrt(2)] * sin(x/2)  or -[sqrt(2)] * sin(x/2)

integral sqrt(1-cos(ax)) dx =integral{ [sqrt(2)] * sin(x/2) }dx 

                                                 = -2 *sqrt(2)*cos(ax/2) +C       or                                             =2*sqrt(2)*cos(ax/2) +C   (use both sines + and -)