Question

\[\int \left( \log x \right)^2 \cdot x\ dx\]

Answer 1

\[\int \left( \log x \right)_{}^2 {x \cdot}    dx\]
`    "Taking  log x"^2" as the first function and x as the second function ". `
\[ = \left( \log x \right)^2 \int xdx - \int\left\{ \frac{d}{dx} \left( \log x \right)^2 \int x\ dx \right\}dx\]
\[ = \left( \log x \right)^2 \cdot \frac{x^2}{2} - \int\frac{\left( 2 \log x \right)}{x} \times \frac{x^2}{2} dx\]
\[ = \left( \log x \right)^2 \times \frac{x^2}{2} - \int x_{II} \log x_I dx\]
\[ = \left( \log x \right)^2 \times \frac{x^2}{2} - \left[ \log x \int x\ dx - \int\left\{ \frac{d}{dx}\left( \log x \right)\int x\ dx \right\}dx \right]\]
\[ = \left( \log x \right)^2 \times \frac{x^2}{2} - \left[ \log x \cdot \frac{x^2}{2} - \int\frac{1}{x} \times \frac{x^2}{2}dx \right]\]
\[ = \left( \log x \right)^2 \times \frac{x^2}{2} - \log x \cdot \frac{x^2}{2} + \frac{x^2}{4} + C\]
\[ = \frac{x^2}{2}\left[ \left( \log x \right)^2 - \log x + \frac{1}{2} \right] + C\]