Mathematics

Evaluate: $$\displaystyle\int \log \left( {\log x} \right) + {\left( {\log x} \right)^{ - 2}}= ?$$


SOLUTION

$$I = \displaystyle \int {\left[ {\log \left( {\log x} \right) + {1 \over {{{\left( {\log x} \right)}^2}}}} \right]dx} $$


                $$ = \displaystyle \int {\log \mathop {\left( {\log x} \right)}\limits_I } .\mathop {1dx}\limits_{II}  + \displaystyle \int {{1 \over {{{\left( {\log x} \right)}^2}}}dx} $$


$$ = \log \left( {\log x} \right).x - \displaystyle \int {{1 \over {\log x}} \times {1 \over x}} .x + \displaystyle \int {{1 \over {{{\left( {\log x} \right)}^2}}}dx} $$


                $$ = \log \left( {\log x} \right) - \displaystyle \int {\mathop {{1 \over {\log x}}}\limits_I .} \mathop {1dx}\limits_{II}  + \displaystyle \int {{1 \over {{{\left( {\log x} \right)}^2}}}dx} $$


$$ = \log \left( {\log x} \right) - {1 \over {\log x}}.x + \displaystyle \int { - {1 \over {{{\left( {\log x} \right)}^2}}}.1dx + } \displaystyle \int {{1 \over {{{\left( {\log x} \right)}^2}}}dx} $$


                $$ = x\log \left( {\log x} \right) - {x \over {\log x}} + c$$  

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Subjective Medium Published on 17th 09, 2020
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