Let u=x2, then du/dx=2x, xdx=½du; xdx/(x4+1)=(½du)/(u2+1).
Let u=tanθ, du/dθ=sec2θ, u2+1=sec2θ, so (½du)/(u2+1)=(½sec2θdθ)/sec2θ=½dθ.
∫xdx/(x4+1)=½∫dθ=θ/2+C (where C is the integration constant)=
½tan-1(u)+C=½tan-1(x2)+C.
CHECK
Differentiate ½tan-1(x2)+C:
Let y=½tan-1(x2)+C, tan(2y-2C)=x2, 1+tan2(2y-2C)=sec2(2y-2C)=1+x4
2sec2(2y-2C)dy/dx=2x, sec2(2y-2C)dy/dx=x, dy/dx=x/(1+x4) which is the original integrand.