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 \( \displaystyle \int_{0}^{\alpha} \frac{t^{50}}{1-t}\,dt \) is equal to:
