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They are called the arguments of the function.
The indefinite integral is the anti-derivative - so the question is, "What function has this given function as a derivative". And if you add a constant to a function, the derivative of the function doesn't change. Thus, for example, if the derivative is y' = 2x, the original function might be y = x squared. However, any function of the form y = x squared + c (for any constant c) also has the SAME derivative (2x in this case). Therefore, to completely specify all possible solutions, this constant should be added.
The domain is a subset of the values for which the function is defined. The range is the set of values that the function takes as the argument of the function takes all the values in the domain.
That set is called the ranger of the function.
domain