Find: [ ( x) + 1 ( x)^ 2 ]dx - Vidyadip

Question

Find: $\int \bigg[\log (\log x) + \frac{1}{(\log x)^{2}}\bigg]dx$

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Answer

$\text{I} = \int\bigg[\log(\log x) + \frac{1}{(\log x)^{2}} \bigg] dx$
$ = \int\log(\log x). 1 dx + \int\frac{1}{(\log x)^{2}}dx$
$= \log(\log x) x - \int\frac{1}{\log x}.\frac{1}{x}.x dx + \int\frac{1}{(\log x)}dx$
$ = x \log(\log x) - \bigg[\frac{1}{\log x}.x - \int\frac{-1}{(\log x)^{2}}.\frac{1}{x}. x \text{ } dx\bigg] + \frac{1}{(\log x)^{2}}dx$
$= x \log (\log x) - \frac{x}{\log x}+\text{C}$

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