Question

$\int {\left(\log x\right)}^2·xdx$

Answer 1

$$\begin{gathered} \int {\underset{}{\left(\log x\right)}}^2\underset{}{x·}dx \\ \mathrm{Taking}{\left(\log x\right)}^2\mathrm{as}\mathrm{the}\mathrm{first}\mathrm{function}\mathrm{and}x\mathrm{as}\mathrm{the}\mathrm{second}\mathrm{function}. \\ ={\left(\log \mathrm{x}\right)}^2\int x\mathit{}dx-\int \left\{\frac{d}{dx}{\left(\log x\right)}^2\int \mathrm{x}dx\right\}dx \\ ={\left(\log x\right)}^2·\frac{x^2}{2}-\int \frac{\left(2\log x\right)}{x}\times \frac{x^2}{2}dx \\ ={\left(\log x\right)}^2\times \frac{x^2}{2}-\int \underset{\mathrm{II}}{x}\underset{\mathrm{I}}{\log x}dx \\ ={\left(\log x\right)}^2\times \frac{x^2}{2}-\left[\log x\int xdx-\int \left\{\frac{d}{dx}\left(\log x\right)\int xdx\right\}dx\right] \\ ={\left(\log x\right)}^2\times \frac{x^2}{2}-\left[\log x·\frac{x^2}{2}-\int \frac{1}{x}\times \frac{x^2}{2}dx\right] \\ ={\left(\log x\right)}^2\times \frac{x^2}{2}-\log x·\frac{x^2}{2}+\frac{x^2}{4}+C \\ \mathit{=}\frac{x^{\mathit{2}}}{\mathit{2}}\left[{\left(\log x\right)}^{\mathit{2}}\mathit{-}\log \mathit{}x+\frac{1}{2}\mathit{}\right]+C \end{gathered}$$