Mathematics

$$\displaystyle \int_{e^{e^{e}}}^{e^{e^{e^{e}}}}\frac{dx}{xlnx\cdot ln\left ( lnx \right )\cdot ln\left ( ln\left ( lnx \right ) \right )}$$ equals


ANSWER

1


SOLUTION
Let $$\displaystyle I=\int _{ { e }_{ { e }_{ e } } }^{ { e }^{ { e }^{ { e }^{ e } } } }{ \frac { 1 }{ x\log { x } .\log { \left( \log { x }  \right) . } \log { \left( \log { \left( \log { x }  \right)  }  \right)  }  }  } dx$$

Substitute $$\displaystyle \log { \left( \log { x }  \right)  } =t\Rightarrow \frac { 1 }{ x\log { x }  } dx=dt$$

$$\displaystyle I=\int _{ e }^{ { { e }^{ e } } }{ \frac { 1 }{ t\log { t }  } dt } =\left[ \log { \log { t }  }  \right] _{ e }^{ { e }^{ e } }=\left[ 1-0 \right] =1$$
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Single Correct Hard Published on 17th 09, 2020
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