In mathematics, there are two types of Euler integral:[1]
For positive integers m and n, the two integrals can be expressed in terms of factorials and binomial coefficients: B ( n , m ) = ( n − 1 ) ! ( m − 1 ) ! ( n + m − 1 ) ! = n + m n m ( n + m n ) = ( 1 n + 1 m ) 1 ( n + m n ) {\displaystyle \mathrm {B} (n,m)={\frac {(n-1)!(m-1)!}{(n+m-1)!}}={\frac {n+m}{nm{\binom {n+m}{n}}}}=\left({\frac {1}{n}}+{\frac {1}{m}}\right){\frac {1}{\binom {n+m}{n}}}} Γ ( n ) = ( n − 1 ) ! {\displaystyle \Gamma (n)=(n-1)!}