Thank you for reporting, we will resolve it shortly
Q.
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 $\int\limits_0^\alpha \frac{t^{50}}{1-t} d t$ is equal to
$ \int\limits_0^\alpha \frac{t^{50}-1+1}{1-t}=-\int\limits_0^\alpha\left(1+ t +\ldots . .+ t ^{49}\right)+\int\limits_0^\alpha \frac{1}{1- t } dt $
$ =-\left(\frac{\alpha^{50}}{50}+\frac{\alpha^{49}}{49}+\ldots . .+\frac{\alpha^1}{1}\right)+\left(\frac{\ln (1- f )}{-1}\right)_0^\alpha$
$ =- P _{50}(\alpha)-\ln (1-\alpha)$
$ =- P _{50}(\alpha)-\beta$