Question

$f$ the graph of the antiderivative $F\left(x\right)$ of $f\left(x\right)=\log \left(\log x\right)+{\left(\log x\right)}^{-2}$
passes through $(e,1998-e)$ then the term independent of $x$ in $F\left(x\right)$ is

Answer 1

Antiderivative of $f\left(x\right)=F\left(x\right)=\int (\log \left(\log x\right)+{\left(\log x\right)}^{-2})dx+C$
$=x\log \left(\log x\right)-\int \frac{x}{x\log x}dx+\int {\left(\log x\right)}^{-2}dx+C=x\log \left(\log x\right)-[x{\left(\log x\right)}^{-1}+\int {\left(\log x\right)}^{-2}dx]+\int {\left(\log x\right)}^{-2}dx+C=x\log \left(\log x\right)-x{\left(\log x\right)}^{-1}+C$
For point $(e,1998-e)$, we get
$1998-e=e.0-e+C\Rightarrow C=1998$