\int (dx)/(xln^(p)x ) How would i integrate this? I tried integration by parts but I couldn't figure out a useful way to use it

Question

`\int (dx)/(xln^(p)x )` How would i integrate this? I tried integration by parts but I couldn't figure out a useful way to use it

Answer 1

Substitute
`u=\ln x`, so that
`\mathrm du=\frac{\mathrm dx}x`. The integral is then equivalent to


`\displaystyle\int(\mathrm dx)/(x\ln^px)=\int(\mathrm du)/(u^p)=\begin{cases}(u^(p+1))/(p+1)+C&\text{for }p\\eq1\\\\\ln|u|+C&\text{for }p=1\end{cases}`

Then transforming back to
`x` gives


`\displaystyle\int(\mathrm dx)/(x\ln^px)=\begin{cases}(\ln^(p+1)x)/(p+1)+C&\text{for }p\\eq1\\\\\ln|\ln x|+C&\text{for }p=1\end{cases}`