jmc

Let the two integers be $x$ and $y$. We are given that $x^2+y^2=193$ and that $xy=84$. We want to find $x+y$. Note that $(x+y{)}^2=x^2+y^2+2xy=193+2\cdot 84=361$. Taking the square root of 361, we see that $x+y=\boxed{19}$.