Indefinite integral

[loud dune]

Hi idk know to do this please help

[wooden stirrup]

remember the logarithm property:
$\ln(x^{n})=n\cdot\ln(x)$

[edgy oracle]

The Impostor

[wooden stirrup]

so if you do the integration, you will get:

$\int\frac{dx}{x\ln(x^{3})}$
$= \int\frac{dx}{3x\ln(x)}$
$= \frac{1}{3}\int\frac{dx}{x\ln(x)}$
$= \frac{1}{3}\int\frac{dx}{v}$
$= \frac{1}{3}\ln|v|+C$
$= \frac{1}{3}\ln(\ln|x|)+C$

(i spent too much time doing the latex code lol)

[edgy oracle]

The Impostor

[wooden stirrup]

note that in the steps, $v = \ln(x)$ so that $dv = \frac{1}{x} dx$

[edgy oracle]

The Impostor

[wooden stirrup]

so i used u-sub 2 times

(tbh idfk my answer is right or not, symbolab and integral-calculator's answers are not the same, but i wrote integral-calculator's version)

[loud dune]

It's wrong🥲 I appreciate your help though!