Double Integrals · Superchallenge problem

For real $u$ let $\{u\} = u - \lfloor u\rfloor$ denote the fractional part. Evaluate $\displaystyle\iint_R \left\{\dfrac{1}{x + y}\right\} dA$, where $R$ is the unit square $0 < x \le 1$, $0 < y \le 1$. You may use $\displaystyle\sum_{n=1}^\infty\dfrac1{n^2} = \dfrac{\pi^2}{6}$. Give an exact answer.