R[x_]:={{x^2*(1-Log[x]),-x^2*Log[x]},{x*Log[x],x*(1+Log[x])}}

MatrixForm[Inverse[R[x]]]

G[x_,t_]:=R[x].Inverse[R[t]]

MatrixForm[FullSimplify[G[x,t]]]

FullSimplify[Integrate[G[x,t].{1,t^2},{t,1,x}]]