Question Number 184497 by Noorzai last updated on 07/Jan/23
Answered by SEKRET last updated on 07/Jan/23
$$\:\:\:\:\:\boldsymbol{\mathrm{x}}^{\boldsymbol{\mathrm{x}}^{\boldsymbol{\mathrm{x}}^{....} } } =\:\:\boldsymbol{\mathrm{y}}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\boldsymbol{\mathrm{x}}^{\boldsymbol{\mathrm{y}}} \:\:=\:\:\boldsymbol{\mathrm{y}} \\ $$$$\:\:\:\boldsymbol{\mathrm{x}}\:=\:\:\boldsymbol{\mathrm{y}}^{\:\frac{\mathrm{1}}{\boldsymbol{\mathrm{y}}}} \:\:\:\:\:\:\:\:\:\:\:\boldsymbol{\mathrm{dx}}=\:−\boldsymbol{\mathrm{y}}^{\frac{\mathrm{1}}{\boldsymbol{\mathrm{y}}}\:−\mathrm{2}} \centerdot\left(\boldsymbol{\mathrm{ln}}\left(\boldsymbol{\mathrm{y}}\right)−\mathrm{1}\right)\boldsymbol{\mathrm{dy}} \\ $$$$\:\:\int\:\boldsymbol{\mathrm{y}}\centerdot\left(−\mathrm{1}\right)\centerdot\boldsymbol{\mathrm{y}}^{\frac{\mathrm{1}}{\boldsymbol{\mathrm{y}}}\:−\mathrm{2}} \left(\boldsymbol{\mathrm{lny}}−\mathrm{1}\right)\:\boldsymbol{\mathrm{dy}}= \\ $$$$\:\:\:−\mathrm{1}\centerdot\int\:\boldsymbol{\mathrm{y}}^{\frac{\mathrm{1}}{\boldsymbol{\mathrm{y}}}\:−\mathrm{1}} \centerdot\left(\:\boldsymbol{\mathrm{ln}}\left(\boldsymbol{\mathrm{y}}\right)\:−\:\mathrm{1}\:\right)\:\boldsymbol{\mathrm{dy}}= \\ $$$$\:=\:\int\:\boldsymbol{\mathrm{y}}^{\frac{\mathrm{1}}{\boldsymbol{\mathrm{y}}}\:−\mathrm{1}} \boldsymbol{\mathrm{dy}}\:−\:\int\boldsymbol{\mathrm{y}}^{\frac{\mathrm{1}}{\boldsymbol{\mathrm{y}}}−\mathrm{1}} \centerdot\boldsymbol{\mathrm{ln}}\left(\boldsymbol{\mathrm{y}}\right)\:\boldsymbol{\mathrm{dy}} \\ $$$$...... \\ $$$$ \\ $$$$ \\ $$