Need to know if the given du value for this integral/ln solution is correct

Question

$\int \frac{2}{x(\ln x+5)^7} dx$ for that problem the example gives $u=\ln x+5$ and $du=\frac{1}{x}dx$. based on a couple other problems I thought du should be $\frac{1}{x+5}dx$ The derivative of inside ln, over what's in ln left alone. Is there something I'm missing about this? Or, since it's happened quite a few times, is this written incorrectly?

Answer 1

Note that inside the parenthesis you have

$$\begin{align*} u &= \ln x + 5\\ &=\ln(x) +5\\ \end{align*} $$

So $$du = \frac 1x \ dx$$

They omitted the parenthesis for the log. Note that $\ln(x+5)$ would never be written as $\ln x + 5$.