Is there a way to compute 
$$I=\int\_{\frac{1-p}{2}}^{\frac{1+p}{2}}x^k(1-x)^kdx$$
for very large positive integer $k\gg 1$ and small $p\in (0,1)$? (one may assume $p$ satisfies $p=\Omega(1/\sqrt{k})$)

I am trying to find a better upper bound on this integral than $p2^{-2k}$. Especially, how far it is from the lower bound $p\Big(\frac{1-p}{2}\Big)^k\Big(\frac{1+p}{2}\Big)^k$?

**PS.**
Is it possible to bound by $2^{-2k}\exp(-\Omega(k^{1/2}))$?