Is Integer Immutable

Immutable does not mean that a can never equal another value. For example, String is immutable too, but I can still do this:

String str = "hello";
// str equals "hello"
str = str + "world";
// now str equals "helloworld"

str was not changed, rather str is now a completely newly instantiated object, just as your Integer is. So the value of a did not mutate, but it was replaced with a completely new object, i.e. new Integer(6).

a is a "reference" to some Integer(3), your shorthand a+=b really means do this:

a = new Integer(3 + 3)

So no, Integers are not mutable, but the variables that point to them are*.

*It's possible to have immutable variables, these are denoted by the keyword final, which means that the reference may not change.

final Integer a = 3;
final Integer b = 3;
a += b; // compile error, the variable `a` is immutable, too.

You can determine that the object has changed using System.identityHashCode() (A better way is to use plain == however its not as obvious that the reference rather than the value has changed)

Integer a = 3;
System.out.println("before a +=3; a="+a+" id="+Integer.toHexString(System.identityHashCode(a)));
a += 3;
System.out.println("after a +=3; a="+a+" id="+Integer.toHexString(System.identityHashCode(a)));

prints

before a +=3; a=3 id=70f9f9d8
after a +=3; a=6 id=2b820dda

You can see the underlying "id" of the object a refers to has changed.

To the initial question asked,

Integer a=3;
Integer b=3;
a+=b;
System.out.println(a);

Integer is immutable, so what has happened above is 'a' has changed to a new reference of value 6. The initial value 3 is left with no reference in the memory (it has not been changed), so it can be garbage collected.

If this happens to a String, it will keep in the pool (in PermGen space) for longer period than the Integers as it expects to have references.

Yes Integer is immutable.

A is a reference which points to an object. When you run a += 3, that reassigns A to reference a new Integer object, with a different value.

You never modified the original object, rather you pointed the reference to a different object.

Read about the difference between objects and references here.