$\begin{align}  & \int_{0}^{2}{\frac{{{(\log x)}^{n}}}{x}dx} \\ & \text{Let }u=\log x \\ & \text{    }\frac{du}{dx}=\frac{1}{x},\text{ }dx=xdu \\ & \int_{0}^{2}{\frac{{{(\log x)}^{n}}}{x}dx}=\int_{0}^{2}{\frac{{{u}^{n}}}{x}(xdu)}=\int_{0}^{2}{{{u}^{n}}du} \\ & \int_{0}^{2}{\frac{{{(\log x)}^{n}}}{x}}=\left[ \frac{{{u}^{n+1}}}{n+1} \right]_{0}^{2}=\left[ \frac{{{(\log x)}^{n+1}}}{n+1} \right]_{0}^{2} \\ & \int_{0}^{2}{\frac{{{(\log x)}^{n}}}{x}dx}=\frac{{{(\log 2)}^{n+1}}}{n+1}-\frac{{{(\log 0)}^{n+1}}}{n+1} \\\end{align}$