Hilbert's tenth problem ask for the general algorithm(finite number of operation) to solve of all Diophantine problems.Today, it is known that no such algorithm exists in the general case.
What about all the solvable general integration problems either in closed form or not(in here, i mean it is not the undefined integral) ? Have any mathematician published a paper about this?
For a good survey of undecidable problems, please see this article by Poonen.
In particular, starting on page 14, there is a discussion of various undecidable problems in analysis. For example, there is no algorithm for determining whether a system of algebraic differential equations (or a single partial differential equation) has a solution. There are early papers on this by Richardson and by Adler, both unfortunately behind pay walls.
In the positive direction, the following result needs to be stated carefully, but roughly speaking there is an algorithm (the Risch Algorithm) for determining whether an elementary function has an elementary antiderivative.