I've come across the following in a text, and can't seem to reproduce it. Can anyone see if, and why, it is true? And if so is it the case for any $f(s',s)$?
$\int_{-t}^tds\int_{-t}^tds'f(s',s)=\frac{1}{2}\int_0^{2t}dx_2\int_{-2t+x_2}^{2t-x_2}dx_1\left[f(x_1,x_2)+f(x_1,-x_2)\right]$
where $x_2=s+s'$ and $x_1=s-s'$.
If it's helpful, the function for which this is being used is
$f(s,s')=e^{-\frac{\gamma^2}{8}(s+s')^2+\frac{i}{2}\omega(s+s')} $.