Integrate $\int \frac{dx}{x(\ln(x))^5}$ using $u$ substitution and the log identity for integrating $x^{-1}$

Question

Section 5.3

Integrate $$\int \frac{dx}{x(\ln(x))^5}$$ using $u$ substitution.

Can somebody verify the following solution for me? Thanks!!

---

Let $u = \ln(x)$. Then $\frac{du}{dx} = \frac{1}{x}$ and thus $x \, du=dx$. Thus we have: \begin{align} \int \frac{dx}{x(\ln(x))^5} &= \int \frac{1}{xu^5} \, x \, du = \int \frac{du}{u^5} = \int u^{-5} du \\ &= \frac{u^{-4}}{-4} + C \\ &= - \frac{1}{4 \ (\ln(x))^4} + C. \end{align}

Answer 1

The old saying is never say never, but I'll be bold and say never, ever mix the variables in a substitution. You should write $$ \int \frac{dx}{x( \ln x)^5}\, dx = \int \frac{1}{(\ln x)^5} \, \frac{dx}{x} = \int \frac 1{u^5} \, du$$ which is exactly what you obtained except for that terrible horrible no good very bad first step.