Calculating steady state response from displacement variance OR finding value of integrand of definite integration from its solution

I am working on optimization of Tuned Mass Damper Inerter (a device used in earthquake engineering). I have calculated variance of displacement and want to calculate steasy state responce using its equation.

Actually it is a mathematics related problem which i want to solve with use of MATLAB. definite integration of response over 0 to inf. is Variance. So i want to go in reverse direction. i.e. want to find response from variance. What should the approach? Please check the attached image.

[Komal Rajana]

Hello, the time-domain response can be obtained via the transfer-function (H(w)) in a simple simulink model.