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Question Number 113865 by Ar Brandon last updated on 15/Sep/20

$$\mathrm{Consider}\mathrm{the}\mathrm{series}{\mathrm{I}}_{\mathrm{n}}={\int }_1^{\mathrm{e}}\mathrm{x}{\left(\mathrm{lnx}\right)}^{\mathrm{n}}\mathrm{dx}\mathrm{and}{\mathrm{I}}_0={\int }_1^{\mathrm{e}}\mathrm{xdx}$$$$\mathrm{Which}\mathrm{of}\mathrm{the}\mathrm{following}\mathrm{is}\mathrm{true}?$$$$\mathrm{a}\mathrm{\setminus }0⩽{\mathrm{I}}_{\mathrm{n}}⩽\frac{{\mathrm{e}}^2}{\mathrm{n}+2}\mathrm{b}\mathrm{\setminus }1⩽{\mathrm{I}}_{\mathrm{n}}⩽\frac{{\mathrm{e}}^2}{\mathrm{n}+1}\mathrm{c}\mathrm{\setminus }{\mathrm{I}}_{\mathrm{n}}\mathrm{is}\mathrm{negative}$$

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