Question
Integrate the following functions w.r.t. x:
$\frac{1}{x \cdot \log x \cdot \log (\log x)}$
$\frac{1}{x \cdot \log x \cdot \log (\log x)}$
$=\int \frac{1}{\log (\log x)} \cdot \frac{1}{x \cdot \log x} d x$
Put $\log (\log x)=t \quad \therefore \frac{1}{\log x} \cdot \frac{1}{x} d x=d t$
$\therefore \frac{1}{x \cdot \log x} d x=d t$
$\begin{aligned} \therefore I & =\int \frac{1}{t} d t=\log |t|+c \\ & =\log |\log (\log x)|+c .\end{aligned}$
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Question is modified
If $|\mathrm{x}|<1$, then prove that $2 \tan ^{-1} \mathrm{x}=\tan ^{-1}\left(\frac{2 x}{1-x^2}\right)=\sin ^{-1}\left(\frac{2 x}{1+x^2}\right)=\cos ^{-1}\left(\frac{1-x^2}{1+x^2}\right)$