The solution for f[x,t] contains a symbolic integral because, basically, the equation is of the form y'[x] == h[x], whose solutions are the primitives of h[x]. If you know nothing about h[x] you must leave the symbolic integration as it is. If you know Xi[1], for example, you can replace it into the solution formula this way:
DSolve[EQ == 0, f[x, t], {x, t}] /.
{Xi[1] -> Function[{x, t}, x + t]} //
Activate
As for the failed solutions with respect to Xi[1] and Xi[2], Mathematica gives back the input because it cannot find a symbolic solution, I suppose.
