Question

n^{p}x }" alt=" \int \frac{dx}{xln^{p}x }" align="absmiddle" class="latex-formula"> 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 $\mathrm du=\dfrac{\mathrm dx}x$. The integral is then equivalent to

$\displaystyle\int\frac{\mathrm dx}{x\ln^px}=\int\frac{\mathrm du}{u^p}=\begin{cases}\dfrac{u^{p+1}}{p+1}+C&\text{for }p\neq1\\\\\ln|u|+C&\text{for }p=1\end{cases}$

Then transforming back to $x$ gives

$\displaystyle\int\frac{\mathrm dx}{x\ln^px}=\begin{cases}\dfrac{\ln^{p+1}x}{p+1}+C&\text{for }p\neq1\\\\\ln|\ln x|+C&\text{for }p=1\end{cases}$