x=(tanθ)√2, dx=(sec2θ)√2dθ, x2+2=2sec2θ.
Integrand 2dx/(x2+2)2=
(2/(4sec4θ))(√2(sec2θ)dθ=(1/√2)cos2θdθ=(1/√2)(½(cos(2θ)+1))dθ.
Integrating: (1/(2√2))(sin(2θ)/2+θ)=(1/(2√2))(sinθcosθ+θ)+C.
If tanθ=x/√2, sinθ=x/√(x2+2), cosθ=√2/√(x2+2), sinθcosθ=x√2/(x2+2).
∫2dx/(x2+2)2=(1/(2√2))(x√2/(x2+2)+tan-1(x/√2))+C=½x/(x2+2)+(1/(2√2))tan-1(x/√2)+C.