How can randomly generated input for a function be used for testing in a systematic way?

How can a method be tested with random input in a systematic way? For example the function int[] foo(int[] data) and it can be checked if the return value is correct or not, after writing test cases for "hard coded inputs" (e.g. {1,2,3}) how can random input be tested? If it's random each time, any errors wont be reproducible. I could print the random input to the screen, but that would become messy for each function. Any suggestions?

Also, is it still unit testing if the whole program is being tested? For example calling the constructor and all methods of class in one @Test? The class only has one public method to begin with so this can't really be avoided, but I'm wondering if JUnit is the best tool for it.