Question

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

• A. 1998
• B. 1999
• C. 1997
• D. 1996

Answer 1

The correct option is A 1998

An antiderivative of $f\left(x\right)=F\left(x\right)$
$=\int \left(log\right(logx)+(logx{)}^{-2}dx+C$ (intergrating by parts the first term)
$=xlog\left(logx\right)-\left[x\right(logx{)}^{-1}+\int \left(logx{)}^{-2}dx\right]+\int (logx{)}^{-2}dx+C$ (again integrating by parts)
$=xlog\left(logx\right)-x(logx{)}^{-1}+C$
Putting $x=e$, we have $1998-e=e.0+e+C.$ Thus $C=1998$.