Question

\[\int\frac{\log x}{x^3} \text{ dx }\]

Answer 1

\[\int\frac{\log x}{x^3}dx\]
\[ = \int \frac{1}{x^3}_{II}\ \log x_I\ dx\]
\[ = \log x\int\frac{1}{x^3}dx - \int\left\{ \frac{d}{dx}\left( \log x \right)\int\frac{1}{x^3}dx \right\}dx\]
\[ = \log x\int x^{- 3} dx - \int\frac{1}{x} \times \left( \frac{x^{- 3 + 1}}{- 3 + 1} \right)dx\]
\[ = \log x \left[ \frac{x^{- 3 + 1}}{- 3 + 1} \right] + \frac{1}{2}\int\frac{1}{x^3}dx\]
\[ = \log x \left( - \frac{1}{2 x^2} \right) + \frac{1}{2}\int x^{- 3} dx\]
\[ = \log x \left( - \frac{1}{2 x^2} \right) + \frac{1}{2} \left[ \frac{x^{- 3 + 1}}{- 3 + 1} \right] + C\]
\[ = \log x \left( - \frac{1}{2 x^2} \right) - \frac{1}{4 x^2} + C\]
\[ = - \frac{1}{4 x^2} \left( 2 \log x + 1 \right) + C\]