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Find The Integral
$\displaystyle \int \bigg( \log( \log x) + \frac{1}{ (\log {x})^{2} } \bigg) dx$
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Answer 1

Step-by-step explanation:

$\huge\mathfrak\orange{answer - > }$

Given:-

• $\displaystyle \int \bigg( \log( \log x) + \frac{1}{ (\log {x})^{2} } \bigg) dx$

☘Solution:-

$I \: = \: ∫ [ log( logx) + \frac{1}{(log \: x) {}^{2} }  ] dx$

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$= \: ∫ \: log \: (log \: x) \: . \: 1 \: dx \: + \: ∫ \frac{1}{(log \: x {}^{2}) } dx$

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$Using \: \: integrating \: \: by \: \: parts \: \: in \: \: first \\ integral, \: \: we \: \: get$

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$⇛I \: \: log \: (log x) \: ∫ \: 1 \: dx \: - \: ∫ [ \frac{d}{dx} \: log( log \: x) \: ∫ 1 \: dx ] \: dx \: + \: ∫ \: \frac{1}{(log \: x {}^{2}) } \: dx$

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$⇛ \: log \: (log \: x.) \: x \: - \: ∫ \: \frac{1}{log \: x} \: . \: \frac{1}{x \: } . \: x \: dx \: + \: ∫ \: \frac{1}{(log \: x {}^{2}) } \: dx$

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$⇛\: x \: log \: (log \: x) \: - \: ∫ (log \: x) {}^{ - 1} \: 1 \: dx \: + \: ∫ \frac{1}{log \: x {}^{2} } \: dx$

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$Again, \: \: applying \: \: integration \: \: by \: \: parts \: \: in \\ the \: \: middle \: \: intergal, \: \: we \: \: get$

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$⇛I \: \: x \: log \: (log \: x) \: - \: [ \: (log \: x) {}^{ - 1} \: \: ∫ 1dx \: - \: ∫ \: \: (\frac{d}{dx } \: ( \: log \: x) {}^{ - 1} \: ∫ 1 \: dx \: ) \: dx] \: + \: ∫ \: \frac{1}{(log \: x) {}^{2} } \: dx \: + C$

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$⇛ \: x \: log \: (log \: x) \: - \: [ \: \frac{x}{log \: x} - \: ∫ \: - \: (log \: x) {}^{ - 2} \: . \: \frac{1}{x} \: . \: x \: dx \: ]  \: + \: ∫ \: \frac{1}{(log \: x) {}^{2} } \: dx \: + \: C$

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$⇛ \: x \: log \: (log \: x) \: - \: \frac{x}{(log \: x)} \: - \: ∫ \frac{1}{(log \: x) {}^{2} } \: dx \: + \: ∫ \: \frac{1}{(log \: x) {}^{2} } \: dx \: + \: C$

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$⇛ \: x \: log \: (log \: x) \: - \: \frac{x}{(log \: x)} \: + C$

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☘αղsաҽɾ ⇛⦅x log (log x) - x(log x) + C