Mathematics - Definite Integration Question with Solution | TestHub

Let

$I={\int }_0^{\alpha }\frac{t^{50}}{1-t}dt$

$\Rightarrow I=-{\int }_0^{\alpha }\left(\frac{1-t^{50}-1}{1-t}\right)dt$

$\Rightarrow I=-{\int }_0^{\alpha }\left(\frac{1-t^{50}}{1-t}\right)dt+{\int }_0^{\alpha }\left(\frac{1}{1-t}\right)dt$

$\Rightarrow I=-{\int }_0^{\alpha }\left(1+t+t^2+...+t^{49}\right)dt-{\left[\ln \left(1-t\right)\right]}_0^{\alpha }$

$\Rightarrow I=-{\left[t+\frac{t^2}{2}+\frac{t^3}{3}+...+\frac{t^{50}}{50}\right]}_0^{\alpha }-\ln \left(1-\alpha \right)$

$\Rightarrow I=-\left[\alpha +\frac{{\alpha }^2}{2}+\frac{{\alpha }^3}{3}+...+\frac{{\alpha }^{50}}{50}\right]-\ln \left(1-\alpha \right)$

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