First, we will determine what the square root of 3 and 12 are:

square root of 3=1.73

square root of 12=3.46

Now we will identify all of the integers between 1.73, and 3.46:

integers: 2, 3

Now we can add these integers to get the sum:

2+3=5

Therefore, 5 is the sum of all the integers between square root of 3 and square root of 12.