Question

If $f_n\left(x\right)=\ln \ln \ln \cdots \ln x(\ln$ is repeated $n$ -times), then $\int {\left[x{f}_1\right(x\left)f_2\right(x)\cdots f_n(x\left)\right]}^{-1}dx$ is equal to
(where $C$ is constant of integration)

• A. $f_{n+1}\left(x\right)+C$
• B. $\frac{f_{n+1}\left(x\right)}{n+1}+C$
• C. $n{f}_n\left(x\right)+C$
• D. $\frac{f_n\left(x\right)}{n}+C$

Answer 1

The correct option is A $f_{n+1}\left(x\right)+C$

$\int \left(\frac{1}{x{f}_1\left(x\right)f_2\left(x\right)\cdots \cdots \cdots f_n\left(x\right)}\right)dx$
Let, $f_n\left(x\right)=t$
$\Rightarrow \left(\frac{1}{f_{n-1}\left(x\right)\cdots \cdots f_1\left(x\right)}\right)\times \left(\frac{1}{x}\right)dx=dt$
$\therefore I=\int \left(\frac{1}{t}\right)dt$
$\Rightarrow I=\ln |t|+C$ $=\ln |f_n\left(x\right)|+C$
$\therefore I=f_{n+1}\left(x\right)+C$