I have the following integral equation $$1 = \int_{-L}^{L} K(s-t)f(t)\mathrm{d}t,$$ where we are solving for $F = \int_{-L}^{L}f(s)\mathrm{d}s$.
That is, $f(t)$ is an unknown function that we do not necessarily need to know. If it helps, the Kernel is $K(x) = -2\,{\frac {{x}^{4}+{\epsilon}^{2}{x}^{2}+3/4\,{\epsilon}^{4}}{ \left( {\epsilon}^{2}+{x}^{2} \right) ^{5/2}}}. $ Does anyone have any ideas on how to find $F$?
P.S.: if it makes things simpler, what if we extend the integration limits to the real line, I.e. $[-L,L]\rightarrow[-\infty,\infty]$? Note that $K(x)$ does not have a computable Fourier transform.