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Beta functions obtained by integration

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Compute the incomplete beta function using Gauss–Legendre quadrature. Setting x=1 gives the complete beta function.
\(\normalsize B_{x}(a,b)={\large\int_{\small 0}^{x}}t^{a-1}(1-t)^{b-1}dt\\\)
Function under integration:
f(t,a,b)=
Variables a
,
b
Integration interval (
,
x)
[ Number of splits n
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\(\normalsize Incomplete\ beta\ functions\ B_x(a,b),\ I_x(a,b)\\ (1)\ Incomplete\ beta\ function\\ \hspace{40px}B_{x}(a,b)={\large\int_{\small 0}^{\hspace{25px}x}}t^{a-1}(1-t)^{b-1}dt\\ (2)\ Regularized\ incomplete\ beta\ function\\ \hspace{40px}I_{x}(a,b)={\large\frac{B_{x}(a,b)}{B(a,b)}}\\ (3)\ Complete\ beta\ function\\ \hspace{40px}B(a,b)={\large\int_{\small 0}^{\small 1}}t^{a-1}(1-t)^{b-1}dt\\ \)

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