_2^4 , , [ _x ,2 , , - , , ( _x ,2 ) ^2 n ,2 ] dx = - Vidyadip

Question

$\int\limits_2^4 {\,\,\left[ {{{\log }_x}\,2\,\, - \,\,\frac{{{{\left( {{{\log }_x}\,2} \right)}^2}}}{{\ell n\,2}}} \right]} $ $dx =$

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Answer

a
$\int\limits_2^4 {\left( {\frac{{\ln \,2}}{{\ln x}}\,\, - \,\,\frac{{\ln 2}}{{{{\ln }^2}x}}} \right)} \,\,\,dx$

$if $ $f(x) $ =$\frac{1}{{\ln \,x}}$

$ \Rightarrow \,\,\,x\,f\,'\,(x)\,\, = \,\, - \,\,\frac{1}{{{{\ln }^2}x}}$

$I = ln2 \left( {\frac{x}{{\ln x}}} \right)_2^4$=$\ln \,2\,\,\left[ {\frac{4}{{\ln 4}}\, - \,\frac{2}{{\ln 2}}} \right]$ = $ 0$

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