If one integer is x the other is 2x, so 1/x+1/(2x)=3/10.
Multiply through by 10x: 10+5=3x and 3x=15 making x=5, so the two integers are 5 and 10.
Another way to solve this is to add 1/x and 1/(2x)=3/(2x)=3/10. The 3s cancel: 1/2x=1/10 so 2x=10 and x=5. The other integer is 10.