Euler integrals

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, 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.)

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