Home -> Solved problems -> Calculate the integral of exponentials

## Calculate the integral

### Solution

\begin{aligned} &\int_{0}^{1} e^{e^{e^{e^{e^{e^{e^{e^{e^{e^{x}}}}}}}}}} \quad e^{e^{e^{e^{e^{e^{e^{e^{e^{x}}}}}}}}} \quad e^{e^{e^{e^{e^{e^{e^{e^{x}}}}}}}} e^{e^{e^{e^{e^{e^{e^{x}}}}}}} e^{e^{e^{e^{e^{e^{x}}}}}} e^{e^{e^{e^{e^{x}}}}} e^{e^{e^{e^{x}}}} e^{e^{e^{x}}} e^{e^{x}} e^{x} d x=?\\\\ &\operatorname{Let} e^{u(x)}=e^{e^{e^{e^{e^{e^{e^{e^{e^{e^{x}}}}}}}}}}\\\\ & u(x)=e^{e^{e^{e^{e^{e^{e^{e^{e^{x}}}}}}}}}\\\\&\quad=>\quad\left(e^{u(x)}\right)^{\prime}=e^{e^{e^{e^{e^{e^{e^{e^{e^{e^{x}}}}}}}}}} \quad e^{e^{e^{e^{e^{e^{e^{e^{e^{x}}}}}}}}} \quad e^{e^{e^{e^{e^{e^{e^{e^{x}}}}}}}} e^{e^{e^{e^{e^{e^{e^{x}}}}}}} e^{e^{e^{e^{e^{e^{x}}}}}} e^{e^{e^{e^{e^{x}}}}} e^{e^{e^{e^{x}}}} e^{e^{e^{x}}} e^{e^{x}} e^{x}\\\\ &\int_{0}^{1} e^{e^{e^{e^{e^{e^{e^{e^{e^{e^{x}}}}}}}}}} \quad e^{e^{e^{e^{e^{e^{e^{e^{e^{x}}}}}}}}} \quad e^{e^{e^{e^{e^{e^{e^{e^{x}}}}}}}} e^{e^{e^{e^{e^{e^{e^{x}}}}}}} e^{e^{e^{e^{e^{e^{x}}}}}} e^{e^{e^{e^{e^{x}}}}} e^{e^{e^{e^{x}}}} e^{e^{e^{x}}} e^{e^{x}} e^{x} d x=\\\\ &=\int_{0}^{1}\left(e^{u(x)}\right)^{\prime} d x=\left.e^{u(x)}\right|_{0} ^{1}\\\\&=e^{e^{e^{e^{e^{e^{e^{e^{e^{e^{1}}}}}}}}}}-e^{e^{e^{e^{e^{e^{e^{e^{e^{e^{0}}}}}}}}}}\\\\&=e^{e^{e^{e^{e^{e^{e^{e^{e^{e}}}}}}}}}-e^{e^{e^{e^{e^{e^{e^{e^{e}}}}}}}} \end{aligned}

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