Let \(\alpha \in (0,1)\) and \(\beta ={\log }_{e}(1-\alpha )\). Let \({P}_{n}(x)=x+\frac{{x}^{2}}{2}+\frac{{x}^{3}}{3}+\…

Let \(\alpha \in (0,1)\) and \(\beta ={\log }_{e}(1-\alpha )\).
Let \({P}_{n}(x)=x+\frac{{x}^{2}}{2}+\frac{{x}^{3}}{3}+\ldots ..+\frac{{x}^{n}}{n},x\in (0,1)\).

Then the integral \({\int }_{0}^{\alpha }\frac{{t}^{50}}{1-t}dt\) is equal to

[JEE Main 2023, 31 Jan (Shift 1)]