Explanation

(a)
Given, $\int_{}^{}{\left\lbrack \log{(\log{x) + \frac{1}{\left( \log x \right)^{2}}}} \right\rbrack dx = x\left\lbrack f(x) - g(x) \right\rbrack + c}$
$LHS = \int_{}^{}{\left\lbrack \begin{matrix}
1 \\
II
\end{matrix}.\log{\begin{matrix}
(\log x) \\
I
\end{matrix}{dx}{+ \int_{}^{}\frac{1}{\left( \log x \right)^{2}}}} \right\rbrack dx}$
$= x\log{(\log{x)}} - \int_{}^{}\frac{1}{\log x}dx + \int_{}^{}{\frac{1}{({\log{x)}}^{2}}\ }dx$
$= x\log{(\log{x)}} - \frac{x}{\log x} - \int_{}^{}\frac{1}{({\log{x)}}^{2}}dx + \int_{}^{}\frac{1}{({\log{x)}}^{2}}dx + c$
$= x\left\lbrack \log{(\log{x) - \frac{1}{\log x}}} \right\rbrack + c$
$\therefore f(x) = \log{(\log{x),g(x) = \frac{1}{\log x}}}\$