∫$$dxx(log$$⁡$$x)(log$$⁡log⁡$$x)$$…(log$$⁡log⁡…⏟$$8$$ times $$x) is equal to

Correct option is 3(log⁡log⁡…⏟9 times x)+C

Questions

$\int \frac{dx}{x (\log x) (\log \log x) \dots \underset{8 times}{\underset{⏟}{(\log \log \dots}} x)}$ is equal to

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(log⁡log⁡…⏟8 times x)+C

(log⁡log⁡…⏟7 times x)+C

(log⁡log⁡…⏟9 times x)+C

None of these

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detailed solution

Correct option is C

Let I=∫dxx(log⁡x)(log⁡log⁡x)…(log⁡log⁡…⏟8 times x)put , (log⁡log⁡…⏟8 times x)=t⇒1(log⁡log⁡…⏟7 times x)…(log⁡log⁡x)(log⁡x)xdx=dt∴ I=∫dtt=log⁡t+C=(log⁡log⁡log⁡…⏟9 times x)+C

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∫dxx(log⁡x)(log⁡log⁡x)…(log⁡log⁡…⏟8 times x) is equal to

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$\int \frac{dx}{x (\log x) (\log \log x) \dots \underset{8 times}{\underset{⏟}{(\log \log \dots}} x)}$ is equal to