Suppose I have some quantity that is defined through the following complicated integral: \begin{align} A(x) = \int \delta[ f(x - y) - f(x + y) ] \, \text{d} y, \end{align} where $\delta$ is the Dirac delta function.
Suppose we know $A_\text{nice}(x)$, assuming that $f = f_\text{nice}(q)$ is a sufficiently nice function. However, now let's say that $f(q) \not= f_\text{nice}(q)$ anymore. Luckily we do know that $f(q) = f_\text{nice}(q + \epsilon(q))$, where $\epsilon(q)$ is a small function that represents a deviation in the coordinate $q$ at which $f_\text{nice}$ is to be evaluated.
So now we have \begin{align} A(x) = \int \delta[ f_\text{nice}(x - y + \epsilon(x - y)) - f_\text{nice}(x + y + \epsilon(x + y))\, \text{d} y \end{align}
Is there any nice way to at least approximate a solution to $A(x)$ such that it relates back to $A_\text{nice}(x)$?
I would imaging a possible change of variable would involve \begin{align} q' &= \epsilon(q) + q, \\ \Rightarrow &= dq' = 1 + \frac{\text{d} \epsilon(q)}{\text{d} q}, \end{align} but making use of this seems unruly.
Any thoughts?