How to perform curve fitting when integrals are involved?

[ShannonZ]

problem description: https://math.stackexchange.com/questions/5053708/how-to-perform-curve-fitting-when-integrals-are-involved

and the references: https://www.sciencedirect.com/science/article/pii/S0167732224008493

https://c.coek.info/pdf-measurement-of-emulsion-droplet-sizes-using-pfg-nmr-and-regularization-methods-.html