Question

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

Answer 1

\[\int x^3 \text{ log x  dx }\]
\[\text{Taking log x as the first function and x^3 as the second function} . \]
\[ = \log x\int x^3 dx - \int\left( \frac{d}{dx}\log x\int x^3 dx \right)dx\]
\[ = \left( \log x \right)\frac{x^4}{4} - \int\frac{1}{x}\left( \frac{x^4}{4} \right)dx\]
\[ = \frac{x^4}{x} \log x - \frac{x^4}{16} + C\]