In my research, I need to calculate $\int_a^bx^m(1-x)^ndx$ where $a<b$ are real numbers and $m,n$ are nonnegative integers. Before I derive it myself, is it written somewhere? I am looking for references.
2026-03-28 07:44:06.1774683846
The Beta function but with arbitrary integral bounds
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The incomplete beta function defines: $$B(y;m+1,n+1)=\int_0^yx^m(1-x)^ndx$$ and so your result will just be: $$B(b;m+1,n+1)-B(a;m+1,n+1)$$