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

Question

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

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Answer

$\text{put}\log\text{x = t} \Rightarrow \text{x} = e^{1} \Rightarrow\text{dx} = e^{1}\text{dt}$
$= \int e^{1}\bigg(\log \text{t}\frac{1}{\text{t}^{2}}\bigg)\text{dt}$
$= \int e^{\text{t}}\Bigg[\bigg(\log \text{t} - \frac{1}{\text{t}}\bigg) + \bigg(\frac{1}{\text{t}} +\frac{1}{\text{t}^{2}}\bigg)\Bigg]\text{dt}$
$= e^{\text{t}}\bigg(\log \text{t}- \frac{1}{\text{t}}\bigg) + \text{c}$
$\text{x}\bigg[\log(\log \text{x}) - \frac{1}{\log \text{x}}\bigg] \text{ + c}$

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