Let X = the first integer

Then x+1 is the second integer

x^2 + (x+1)^2 = 113

Expanding the left-hand expression and moving the 113 to the left side:

x^2 + x^2 + 2x + 1 - 113 = 0

2x^2 + 2x - 112 = 0

Simplfying the left side:

x^2 + x - 56 = 0

Factoring:

(x + 8)(x - 7) = 0

So the first integer is 7 or -8

and the second integer is either 8 or -7