#int_0^(2x)sqrt(1+sin(x/2))*dx#
#color(green)(sin(x/2)=2sin(x/4)cos(x/4)#
#color(green)(1=sin^2(x/4)+cos^2(x/4)#
Substitute
#int_0^(2x)sqrt(1+sin(x/2))*dx#
#=int_0^(2x)sqrt(sin^2(x/4)+2sin(x/4)cos(x/4)+cos^2(x/4))*dx#
By Completing The Square
#=int_0^(2x)sqrt((sin(x/4)+cos(x/4))^2)*dx#
#=int_0^(2x)(sin(x/4)+cos(x/4))*dx#
#=[sin(x/4)+cos(x/4)]_0^(2x)#
#=sin(x/2)+cos(x/2)-1#
#color(blue)("if this question was like this " sqrt(1+cos(x/2))
" it can be simplified to look like this" #
#color(blue)(sqrt2*cos(x/4) "using the Double angles Formulae"#
