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

A
$\underset{8 times}{\underset{⏟}{(\log \log \dots}} x) + C$
B
$\underset{7 times}{\underset{⏟}{(\log \log \dots}} x) + C$
C
$\underset{9 times}{\underset{⏟}{(\log \log \dots}} x) + C$
D
None of these

Solution:

$\begin{array}{l} Let I = \int \frac{dx}{x (\log x) (\log \log x) \dots \underset{8 times}{\underset{⏟}{(\log \log \dots}} x)} \\ put, \underset{8 times}{\underset{⏟}{(\log \log \dots}} x) = t \end{array}$

$\begin{array}{l} \Rightarrow \frac{1}{\underset{7 times}{\underset{⏟}{(\log \log \dots}} x) \dots (\log \log x) (\log x) x} dx = dt \\ ∴ I = \int \frac{dt}{t} = \log t + C = \underset{9 times}{\underset{⏟}{(\log \log \log \dots}} x) + C \end{array}$

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

Solution:

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