View source for Euler integrals

{{MSC|33B15}}

The integral
$$
B(p,q) = \int_0^1 x^{p-1}(1-x)^{q-1}\rd x, \quad p,q > 0,
$$
called the Euler integral of the first kind, or the [[Beta-function|beta-function]], and
$$
\int_0^\infty x^{s-1}e^{-x} \rd x,
$$
called the Euler integral of the second kind. (The latter converges for $s>0$ and is a representation of the [[Gamma-function|gamma-function]].)

These integrals were considered by L.&nbsp;Euler (1729–1731).