По частям
u = arctg(x/2); du = 1/2*dx/(1+x^2)
dv = x dx; v = x^2/2
Int = u*v - Int v du = x^2/2*arctg(x/2) - 1/4*Int x^2/(1+x^2) dx =
= x^2/2*arctg(x/2) - 1/4*Int (1 - 1/(1+x^2)) dx = x^2/2*arctg(x/2) - x/4 + 1/4*arctg x + C
