Question

If $f_n\left(x\right)=\log \log \log \dots \log x(\log$ is repeated $n-times),$ then $\int {\left[x{f}_1\right(x\left)f_2\right(x)\dots f_n(x\left)\right]}^{-1}dx=$

Answer 1

$\int \left(\frac{1}{x{f}_1\left(x\right)f_2\left(x\right)..........f_n\left(x\right)}\right)dxlet{f}_n\left(x\right)=tthen\left(\frac{1}{f_{n-1}\left(x\right).......f_1\left(x\right)}\right)\times \left(\frac{1}{x}\right)dx=dt\therefore I=\int \left(\frac{1}{t}\right)dt=logt+C=log{f}_n\left(x\right)+C=f_{n+1}\left(x\right)+C$