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
![=∫_0 ^( 1) e^e^e^x e^e^x d(e^x )=∫_0 ^( 1) e^e^e^x d(e^e^x )=[e^e^e^x ]_0 ^1 =e^e^e −e^e](https://www.tinkutara.com/question/Q210210.png)
$$=\int_{\mathrm{0}} ^{\:\mathrm{1}} {e}^{{e}^{{e}^{{x}} } } {e}^{{e}^{{x}} } {d}\left({e}^{{x}} \right)=\int_{\mathrm{0}} ^{\:\mathrm{1}} {e}^{{e}^{{e}^{{x}} } } {d}\left({e}^{{e}^{{x}} } \right)=\left[{e}^{{e}^{{e}^{{x}} } } \right]_{\mathrm{0}} ^{\mathrm{1}} ={e}^{{e}^{{e}} } −{e}^{{e}} \\ $$
Answered by Spillover last updated on 03/Aug/24
Answered by klipto last updated on 06/Aug/24
![u=e^e^x du=e^x e^e^x dx ∫_0 ^1 e^u du=e^u ]_0 ^1 =(e^e^e^x −e^e^e^x )=e^e^e −e^e ✓](https://www.tinkutara.com/question/Q210323.png)
$$\boldsymbol{\mathrm{u}}=\boldsymbol{\mathrm{e}}^{\boldsymbol{\mathrm{e}}^{\boldsymbol{\mathrm{x}}} } \:\:\:\boldsymbol{\mathrm{du}}=\boldsymbol{\mathrm{e}}^{\boldsymbol{\mathrm{x}}} \boldsymbol{\mathrm{e}}^{\boldsymbol{\mathrm{e}}^{\boldsymbol{\mathrm{x}}} } \boldsymbol{\mathrm{dx}} \\ $$$$\left.\int_{\mathrm{0}} ^{\mathrm{1}} \boldsymbol{\mathrm{e}}^{\boldsymbol{\mathrm{u}}} \boldsymbol{\mathrm{du}}=\boldsymbol{\mathrm{e}}^{\boldsymbol{\mathrm{u}}} \right]_{\mathrm{0}} ^{\mathrm{1}} =\left(\boldsymbol{\mathrm{e}}^{\boldsymbol{\mathrm{e}}^{\boldsymbol{\mathrm{e}}^{\boldsymbol{\mathrm{x}}} } } −\boldsymbol{\mathrm{e}}^{\boldsymbol{\mathrm{e}}^{\boldsymbol{\mathrm{e}}^{\boldsymbol{\mathrm{x}}} } } \right)=\boldsymbol{\mathrm{e}}^{\boldsymbol{\mathrm{e}}^{\boldsymbol{\mathrm{e}}} } −\boldsymbol{\mathrm{e}}^{\boldsymbol{\mathrm{e}}} \checkmark \\ $$
Answered by Spillover last updated on 07/Aug/24
