$U_n= \int\frac{x^n}{((x(1-x))^{0.5}}$ where $0<x<1$
Prove that $2nU_n=(2n-1)U_{n-1}$
My work I did $U_0=\pi, u_1=\pi/2$ so its true for $n=1$
2026-05-05 00:47:33.1777942053
Definite integration by induction
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Hint: You have an integral involving $x^n$, and you would like to rewrite it as an integral involving $x^{n-1}$. Consider integrating by parts.