$\displaystyle \sum_{k=0}^{n} {n\choose{k}}x^{n-k}y^k$
$\displaystyle = x^n+nx^{n-1}y+\frac{n(n-1)}{2!}x^{n-2}y^2+\cdots +\frac{n(n-1)\cdots(n-r+1)}{r!}x{n-r}y^r+\cdots+y^n$
So taking:
$\displaystyle \int\sum_{k=0}^{n}{n\choose{k}}x^{n-k}y^kdx$
Is just a case of using the addition and power rules for integrals on this series.