If the first of the two integers is [tex] x [/tex], the next one is [tex] x+1 [/tex]. So, the sum of their cubes are
[tex] x^3+(x+1)^3 = x^3+x^3 + 3 x^2 + 3 x + 1 = 2x^3+3x^2+3x+1 [/tex]
So, we have the following equation:
[tex] 2x^3+3x^2+3x+1 =1241 \iff 2x^3+3x^2+3x-1240 =0[/tex]
By the rational root theorem, if this polynomial admits a rational root, it is a fraction [tex] \frac{p}{q} [/tex] where p divides 1240 and q divides 2 (i.e., q=1 or q=2). You'll have to go with a bit of trial and error here, because the standard formula for solving cubic equation is quite complicated.
Eventually, you'll arrive to p=8, q=1, and if you plug 8 into the equation you'll see that it is a solution. So, the two numbers are 8 and 9, in fact
[tex] 8^3+9^3=512+729=1241[/tex]
