Let $\alpha \in (0,1)$ and $\beta = {\log _e}(1 - \alpha )$. Let ${P_n}(x) = x + {{{x^2}} \over 2} + {{{x^3}} \over 3}\, + \,...\, + \,{{{x^n}} \over n},x \in (0,1)$. Then the integral $\int\limits_0^\alpha {{{{t^{50}}} \over {1 - t}}dt} $ is equal to

Question

$ - \left( {\beta + {P_{50}}\left( \alpha \right)} \right)$

$\beta - {P_{50}}(\alpha )$

${P_{50}}(\alpha ) - \beta $

$\beta + {P_{50}} - (\alpha )$

Solution

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