Is there a general inversion formula or procedure for an integral of the form (where f is the function being transformed and g depends on the type of transform) $\int^{a}_{b} f(x) g(x,\xi) dx $ ? Inverses are defined in the conventional ways for functionals and integral transforms, respectively.

For instance, for the fourier transform. In the equation above, $a=∞,b=-∞$, $g(x,\xi)=e^{-2πix \xi }$. I know the inverse fourier transform is simple but I am concerned with a general procedure or process.

I am most interested in the cases where $a,b=±∞$ although a simple inverse for $\int^{a}_{b} f(x) dx $ is also something I am curious about (as far as this part of the question is concerned, if the inverse of an indefinite integral is the derivative, what is the inverse of a definite integral-- I am sorry if this is too elementary).

Feel free to use complex analysis or any other branch of math if it helps to answer the question. Also, you can repost on another site if it will help.