As always thank you for taking the time over this.
I had wrongly assumed that the appearance of imaginary parts of any size was a warning that a real system was not being represented. Two consequential questions if I might take up some more of your time.
Firstly, would I be right in thinking that in general that if I set Digits to a higher value and the already small imaginary part gets smaller then that would always be a confirmation that the apperance of complex numbers is a rounding issue?
Secondly, if I using your definitions write
evalf(eval(J1 - eval(J1, x=0) - 1, x = v)) = evalf(Int(1/(sin(x+1/2)+1/10),x=0..v))
am I to understand that mathematically this is correct but that due to limitations in computer solving methods the left hand side produces the wrong answer?
Once again many thanks.