Pdepe - how to compute/possible an integral over space in pdefun(int​egro-diffe​rential)?

This maybe a bit of a naive question but I'm rather new to the argument of both pdes and integration of such in matlab.

I'm using pdepe avialabe in matlab 2018b.

I'm intergrating something that is made by a classic fokker plank part with some additional terms that I have indicated in the following equation as "..." and I am not too concerned about.

I have sucessfully managed to integrate a simple fokker planck equation using pdepe and all is working perfectly(that would be the equation above withouth the terms in lambda and the omitted terms denoted by "...")

However, I would need to integrate also making use of the other terms of the equation.

The term after the dots i.e. lambda hat is described by the following equation:

Hence, it is a integro differential equation where I would need to integrate (specifically calcualte the expectation) of u in order to obtain a solution.

Is there anyway I can handle this in matlab other than using the method of lines and having going trough such hardship?

Thank you in advance.