Beta integral ($B(a,b)={\int }_0^1{x}^{a-1}(1-x{)}^{b-1}dx$)

For positive real numbers $a,b$, the beta integral represents the beta function by $B(a,b)={\int }_0^1{x}^{a-1}(1-x{)}^{b-1}dx$. In particular, for nonnegative integers $m,n$, it gives ${\int }_0^1{x}^m(1-x{)}^{n}dx=m!n!/(m+n+1)!$, connecting integration to binomial coefficients.