If x is the first integer, then
x+1 is the second,
x+2 is the third, and
x+3 is the fourth.
Now we need those to be such that adding the first three equals the fourth plus 18.
x + x+1 + x+2 = x+3 + 18
Solving that equation, we'd do:
3x + 3 = x + 21
2x = 18
x = 9
That means 9 is the first integer, so the others are 10, 11, and 12.