Beta Function as Integral of Power of t over Power of t plus 1

Theorem

 * $\ds \map \Beta {x, y} = \int_{\mathop \to 0}^{\mathop \to \infty} \frac {t^{x - 1} } {\paren {1 + t}^{x + y} } \rd t$

where $\Beta$ denotes the Beta function.

Proof
Consider the substitution $s = \dfrac t {1 + t}$.

We have the following:


 * $\dfrac 1 {1 + t} = 1 - s$
 * $t \to 0, s \to 0$
 * $t \to \infty, s \to 1$
 * $\rd s = \dfrac 1 {\paren {1 + t}^2} \rd t$

Then: