I am in a situation where I want to use mutable versions of things like Integer. Do I have to use these classes (below) or does Java have something built in?
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You can use an nnnn[] as a mutable object for any primitive type as @Alexandre suggests, java also has AtomicInteger and AtomicLong.
IMHO int is usually a better choice than Integer and that is mutable.
Can you more details of why you need a mutliple object, perhaps there is another way to achieve the same thing.