Let $C_n(x) = \frac{n!}{\Gamma(x)\cdot \Gamma(n-x)}$

Is the following true: $\int_{0}^{n} [C_n(x)\cdot y^x \cdot (1-y)^{n-x}dx] = 1$??

just wondering

In generality for continuous functions $f,g$ from the reals to the reals is it the case that:

$\int_{0}^{n} [C_n(x)\cdot f(y)^x \cdot g(z)^{n-x}dx] = [f(y)+g (z)]^{n}$

Another integral: let $f,g : R \mapsto R$

Is $\int_{-\infty}^{\infty}\int_{-y}^{y} f(y-x)g(y+x)dxdy = [\int_{-\infty}^{\infty}f(x)dx][\int_{-\infty}^{\infty}g(x)dx]$