Find the antiderivative of `f(x) = In (In x) + (In x)^-2` whose graph passes through `(e, e)`.

Options

A. `x[log(logx)+(logx)^(-1)]`
B. `x[-log(logx)+(logx)^(-1)]+e`
C. `x[log(logx)-(logx)^(-1)]+2e`
D. None of the above

Answer

C

Explanation

An antiderivative of `f(x)=F(x)`
`=int[log(logx)+(logx)^(-2)]dx+c`
`=xlog(logx)-int(x)/(xlogx)dx+int(logx)^(-2)dx+c`
Integrating by parts in the first integral
`=xlog(logx)-[x(logx)^(-1)+int(logx)^(-2)dx]+int(logx)^(-2)dx+c`
[Again integrating by parts in `(log x)^(-1)`]
`=xlog(logx)-x(logx)^(-1)+c`
Putting x = e, we have `e=0-e+c`
So, `" "c=2e`
Thus,
`f(x)=x[log(logx)-(logx)^(-1)]+2e`