Question

$\int x^3\log xdx=$

Answer 1

$\begin{array}{c}\int x^3\log xdx\\ =\log x\int x^3-\int \left[\frac{d}{dx}\log x\int x^{3}dx\right]\\ =\frac{\log x.x^4}{4}-\int \left[\frac{1}{x}.\frac{x^4}{4}+c\right]\\ =\frac{\log x.x^4}{4}-\frac{1}{4}\int x^{3}dx\\ =\frac{\log x.x^4}{4}-\frac{1}{16}{x}^4+c\\ =\frac{1}{4}{x}^4\left[\log x-\frac{1}{4}\right]+c\end{array}$