If one integer is x then the other is 2x-3.
The sum of their squares is 698 so x²+(2x-3)²=698.
Expanding: x²+4x²-12x+9=698, 5x²-12x-689=0. The factors of 689 include 13*53.
We can factorise: (5x+53)(x-13)=0, so x=13 is one positive integer and the other is 23.
CHECK: 13²+23²=169+529=698.