Question

Evaluate: $\int x^3\log xdx$

Answer 1

Solution:-

$\int x^3\log xdx=\log x\int x^{3}dx-(\int \frac{d\left(\log x\right)}{dx}(\int x^3)dx)$

$\Rightarrow =\log x\left(\frac{x^4}{4}\right)-(\int \frac{1}{x}\left(\frac{x^4}{4}\right)dx)\{\because \frac{d\left(\log x\right)}{dx}=\frac{1}{x}\}$

$\Rightarrow =\log x\frac{x^4}{4}-(\int \frac{x^3}{4}dx)$

$\Rightarrow =\log x\frac{x^4}{4}-\frac{x^4}{4\times 4}$

$\Rightarrow =\frac{x^4}{4}(\log x-\frac{1}{4})$