Algebra Questions

Question Number 210208 by Spillover last updated on 02/Aug/24

$$\\$$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \:{e}^{{e}^{{e}^{{x}} } } \:{e}^{{e}^{{x}} } \:{e}^{{x}} {dx} \\$$$$\\$$

Answered by mahdipoor last updated on 02/Aug/24

$$=\int_{\mathrm{0}} ^{\:\mathrm{1}} {e}^{{e}^{{e}^{{x}} } } {e}^{{e}^{{x}} } {d}\left({e}^{{x}} \right)=\int_{\mathrm{0}} ^{\:\mathrm{1}} {e}^{{e}^{{e}^{{x}} } } {d}\left({e}^{{e}^{{x}} } \right)=\left[{e}^{{e}^{{e}^{{x}} } } \right]_{\mathrm{0}} ^{\mathrm{1}} ={e}^{{e}^{{e}} } −{e}^{{e}} \\$$

Answered by Spillover last updated on 03/Aug/24

Answered by klipto last updated on 06/Aug/24

$$\boldsymbol{\mathrm{u}}=\boldsymbol{\mathrm{e}}^{\boldsymbol{\mathrm{e}}^{\boldsymbol{\mathrm{x}}} } \:\:\:\boldsymbol{\mathrm{du}}=\boldsymbol{\mathrm{e}}^{\boldsymbol{\mathrm{x}}} \boldsymbol{\mathrm{e}}^{\boldsymbol{\mathrm{e}}^{\boldsymbol{\mathrm{x}}} } \boldsymbol{\mathrm{dx}} \\$$$$\left.\int_{\mathrm{0}} ^{\mathrm{1}} \boldsymbol{\mathrm{e}}^{\boldsymbol{\mathrm{u}}} \boldsymbol{\mathrm{du}}=\boldsymbol{\mathrm{e}}^{\boldsymbol{\mathrm{u}}} \right]_{\mathrm{0}} ^{\mathrm{1}} =\left(\boldsymbol{\mathrm{e}}^{\boldsymbol{\mathrm{e}}^{\boldsymbol{\mathrm{e}}^{\boldsymbol{\mathrm{x}}} } } −\boldsymbol{\mathrm{e}}^{\boldsymbol{\mathrm{e}}^{\boldsymbol{\mathrm{e}}^{\boldsymbol{\mathrm{x}}} } } \right)=\boldsymbol{\mathrm{e}}^{\boldsymbol{\mathrm{e}}^{\boldsymbol{\mathrm{e}}} } −\boldsymbol{\mathrm{e}}^{\boldsymbol{\mathrm{e}}} \checkmark \\$$

Answered by Spillover last updated on 07/Aug/24