One idea can lie in the use of a family of functions the decompose the unknown f. Choose an orthogonal family over the chosen domain, so 0 to infinity in this case. That may mean Bessel functions. It may mean something like Laguerre polynomials. That is, we can write the integral in a form like
intt(K*f*dt) = int(K*exp(-t)*exp(t)*sum(a_i*p_i(t))*dt)
that is, if the polynomials P_i are Laguerre polynomials, they will be orthogonal with the inclusion of that exp(-t) in that Kernel. So now you write f(t) as a sum of the form
f(t) = sum(a_i*p_i(t))
This will work if that exp(t) term can be neatly absorbed into K.
How does this help? You will need to use a finite number of terms in that sum of course. But the integral and the sum can exchange places. And now you can try to solve for the unknown constants a_i.
The above are standard solution methods for this class of problems, though it has been many years since I worried about them.
Again, as I said before, you will need to do some reading, or spend some time talking to someone with expertise in these solutions.