Mathematics

$$\int 2^{2^{2^{x}}} 2^{2^{x}} 2^{x}dx$$.


SOLUTION

$$\int { { 2 }^{ { 2 }^{ { 2 }^{ x } } } } { 2 }^{ { 2 }^{ x } }{ 2 }^{ x }\quad dx$$

Let $${ 2 }^{ x }=t\\ \cfrac { dt }{ dx } ={ 2 }^{ x }\ln { 2 } \\ { 2 }^{ x }dx=\cfrac { dt }{ \ln { 2 }  } \\ { 2 }^{ { 2 }^{ t } }{ 2 }^{ t }\cfrac { dt }{ \ln { 2 }  } $$

Let $${ 2 }^{ t }=m\\ \cfrac { dm }{ dt } ={ 2 }^{ t }\ln { 2 } \\ { 2 }^{ t }dt=\cfrac { \left( dm \right)  }{ \ln { 2 }  } \\ \int { \cfrac { { 2 }^{ m }dm }{ { \left( \ln { 2 }  \right)  }^{ 2 } }  } =\cfrac { 2^m }{ { \left( \ln { 2 }  \right)  }^{ 3 } } =\cfrac { { 2 }^{ { 2 }^{ { 2 }^{ x } } } }{ { \left( \ln { 2 }  \right)  }^{ 3 } } +C$$

 

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Subjective Medium Published on 17th 09, 2020
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