Question

What would be the u to usub and what would be the steps to solving this integral?

Answer 1

Presumably, ln⁵(x) is the same as (ln(x))⁵ (as opposed to a quintuply-nested logarithm, log(log(log(log(log(x)))))).

Then substituting u = ln(x) and du = dx/x gives

`\displaystyle\int(\mathrm dx)/(x\ln^5(x)) = \int(\mathrm du)/(u^5) = -\frac1{4u^4}+C = \boxed{-\frac1{4\ln^4(x)}+C}`