I saw a problem similar to this: it was needed to find a estimation for, \begin{align} \int_0^1 \int_0^{1} x^n(1-x)^ny^n(1-y)^n dx dy \end{align} with $n \in \mathbb{Z}$ and $\geq 1$. The resolution goes: define $f(x,y) = x(1-x)y(1-y)$ with $ x,y\in (0,1)$. The maximum of $f$ occurs at $f(1/2,1/2) = 1/16$, therefore, \begin{align} \int_0^1 \int_0^{1} x^n(1-x)^ny^n(1-y)^n dx dy < (1/16)^n \end{align}
I wonder if we can do something similar to estimate: \begin{align} \int_0^1 \int_0^{1-y} x^n(1-x)^ny^n(1-y)^n dx dy \end{align} with $ x,y\in (0,1)$ and $1\leq n \in \mathbb{Z}$, without actually evaluating the integral. I'm not even sure if this makes sense. The integral above is a real number depending on $n$, I want to estimate this value, bound the integral, like the problem before. Also, the integrals are improper ones.