Question

\[\int\left( x + 1 \right) \text{ log  x  dx }\]

Answer 1

\[\int \left( x + 1 \right)_{II} . \log_1 \text{  x  dx }\]
\[ = \log x\int\left( x + 1 \right)dx - \int\left\{ \frac{d}{dx}\left( \log x \right)\int\left( x + 1 \right)dx \right\}dx\]
\[ = \log x\left[ \frac{x^2}{2} + x \right] - \int \frac{1}{x}\left( \frac{x^2}{2} + x \right)dx\]
\[ = \log x\left( \frac{x^2}{2} + x \right) - \int \left( \frac{x}{2} + 1 \right)dx\]
\[ = \log x\left( \frac{x^2}{2} + x \right) - \left( \frac{x^2}{4} + x \right) + C\]