I computed the following integral and got that $$\int_S [x_1y_3+x_2(y_2+y_3)+x_3(y_1+y_2)]^n =O\Big(\frac{\log(n)}{n^3}\Big).$$ The integral is over the set $S=\{x_1+x_2+x_3=1, y_1+y_2+y_3=1 | x_i,y_i\geq 0\}$.
I now want evaluate a somewhat different integral, of the form: $$\int\limits_S \sum\limits_k \binom{n}{k}\frac1k [x_1y_3+x_2(y_2+y_3)+x_3(y_1+y_2)]^{n-k}(x_3y_3)^k$$ over the same domain. So far I have found an upper bound of the form $O\Big(\frac{\log^2(n)}{n^2}\Big)$, which is not good for me. I believe that this upper bound can be improved to $o\Big(\frac1{n^2}\Big)$ but cannot prove this currently.
Is there any good way to evaluate this integral? Any way to prove\disprove that it's $o\Big(\frac1{n^2}\Big)$?