co.combinatorics - Gamma function versions of combinatorial identites?

Chapter 5.5 of Concrete Mathematics discusses generalizing binomial coefficient identities to the Gamma function. It doesn't discuss the two integrals you mention, though.

Doing a bit of thinking on my own, if $n$ is a positive integer then
$$int_{z=0}^n binom{n}{z} dz = int_{z=0}^n frac{n! dz}{Gamma(1+z) Gamma(n+1-z)}$$
$$int_{z=0}^{n} frac{n! dz}{(n-z)(n-1-z) cdots (1-z) Gamma(1-z) Gamma(1+z)}.$$

We have $Gamma(1+z) Gamma(1-z) = pi z/sin (pi z)$, if I haven't made any dumb errors, so this is
$$int_{0}^n frac{ n! sin (pi z) dz}{pi z (n-z)(n-1-z) cdots (1-z)}.$$

I suspect this integrand does not have an elementary anti-derivative, because it reminds me of $int sin t dt/t$. But there might be some special trick which would let you compute the integral between these specific bounds.