How do I evaluate the following integral? $$ \int_{0}^{c} y^{\alpha-1} (1-y)^{\beta-1}dy $$ where $1\geq c>0$.
Thank you in advance.
2026-03-25 11:54:11.1774439651
How to evaluate this$ \int_{0}^{c} y^{\alpha-1}(1-y)^{\beta-1}dy$?
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As already pointed out by Somos within the comment section this is the so-called Incomplete Beta Function denoted as
Further note that for $z=1$ this equals the Beta Function $($as mentioned by Travis$)$
$$B(a,b)=\int_0^1 t^{a-1}(1-t)^{b-1}dt$$
The Beta Function $-$ and the Incomplete Beta Function aswell $-$ are special functions and cannot be represented in terms of elementary functions $($such as polynomials, logarithms, etc.$)$ in a finite combination. So the integral
$$\int_0^c y^{\alpha-1}(1-y)^{\beta-1}dy$$
cannot be "evaluated" in the classical sense as long as the numbers $\alpha,\beta,c$ are choosen arbitrarily. You may look up the two links for further information especially concerning their different ways of representation.