Mathematics

Evaluate the following integral
$$\int { \cfrac { 1 }{ x\log { x } \log { \left( \log { x }  \right)  }  }  } dx\quad $$


SOLUTION
Let $$t=\log{x}\Rightarrow\,dt=\dfrac{1}{x}dx$$

$$\displaystyle\int{\dfrac{dx}{x\log{x}\log{\left(\log{x}\right)}}}$$

$$=\displaystyle\int{\dfrac{dt}{t\log{t}}}$$

Let 
$$u=\log{t}\Rightarrow\,du=\dfrac{1}{t}dt$$

$$=\displaystyle\int{\dfrac{du}{u}}$$

$$=\log{u}+c$$

$$=\log{\log{t}}+c$$ where $$u=\log{t}$$

$$=\log { \left[ \log { \left( \log { x }  \right)  }  \right]  } +C$$ where $$t=\log{x}$$
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Subjective Medium Published on 17th 09, 2020
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