I am trying to use the following mathematical model to model some Physical properties of materials. Alpha and Beta are constants but I have a question about the integration. Here is the equation:
Equation from McIntyre (1966) that I am trying to solve
My question is: there is a dx' (and dx'') in the integrals. Am I supposed to evaluate this integral any differently than normal? Additionally, there is an x' bound. How would this be handled?
as there is nothing else relating to $x$ present in each of the integrals, you can treat $x'$ as a variable in its own right, that is to say: $$\int dx'=x'+C$$ if there were an expression of $x$ present it would be much more difficult although $x$ could also represent a function i.e. $x=x(t)$ which would give you: $$dx'=d\left[\frac{dx}{dt}\right]=\frac{d^2x}{dt^2}dt$$