Hello, Heather!
Last one?
\(\displaystyle \L\int \frac{\ln(x^5)}{x}\.dx\)
Let \(\displaystyle u = \ln(x^5)\;\;\Rightarrow\;\;du = \frac{5x^4}{x^5}\.dx\;\;\Rightarrow\;\;dx\,=\,\frac{x}{5}\,du\)
Substitute: \(\displaystyle \L\;\int\frac{u}{x}\,\left(\frac{x}{5}\,du\right) \;= \;\frac{1}{5}\int u\,du\;=\;\frac{1}{10}u^2\,+\,C\)
Back-substitute: \(\displaystyle \L\;\frac{1}{10}[\ln(x^5)]^2\,+\,C\)
Seeing that you posted so many similar problems,
. . I assumed you are new to (or lost in) Substitution
. . and gave you detailed solutions.
I hope the mystery is cleared up now.
