I'm sorry, your solution is very good and most succinct but to gain full marks for my assignment think i have to use the integral of

x^2 / ( 1+x^2 )

I have, from more directly integrating by parts of x.arctan (x) where

u=arctan(x) giving du= dx / (1+x^2)

dv=x giving v=1/2.x^2

in formula int u.dv = u.v. - int v.du

shows the significance of the x^2 / ( 1+x^2 ) term by -

1/2.x^2.arctan(x) - 1/2. int x^2 / ( 1+x^2 )

I am glad to have seen your method, I have not seen that technique before (your very first step of sub know integral of x^2 + 1, does it have a name?) I have tried to see how it might be relavent to my current position as I have attempted to explain - but failed again unfortunately. If you could again assist i would be most humbled. Also, are you inserting Images made in / with word equation editor?

many thanks !