Beta Function of x with y+1 by x+y over y

Theorem
Let $\Beta \left({x, y}\right)$ denote the Beta function.

Then:
 * $\Beta \left({x, y}\right) = \dfrac {x + y} x \Beta \left({x, y + 1}\right)$

Proof
By definition of Beta function:


 * $\displaystyle \Beta \left({x + 1, y}\right) = \int_0^1 t^x \left({1 - t}\right)^{y - 1} \, \mathrm d t$

With a view to expressing the primitive in the form:
 * $\displaystyle \int u \frac {\mathrm d v}{\mathrm d x} \ \mathrm d x = u v - \int v \frac {\mathrm d u}{\mathrm d x} \ \mathrm d x$

let:

and let:

Then:

Hence: