Question

$I=\int x^3\log xdx$

Answer 1

$\int x^{3}logx\cdot dx$

$\int udv=uv-\int v.du$

$u=logx$; $dv=x^{3}dx$

$du=\frac{1}{x}dx$ $v=\frac{x^4}{4}$

$\int x^{3}logx=\frac{x^4}{4}log\left(x\right)-\int \frac{x^4}{4}\cdot \frac{1}{x}dx$

$=\frac{x^4}{4}logx-\frac{1}{4}\int x^{3}dx$

$=\frac{x^4}{4}logx-\frac{1}{4}\frac{x^4}{4}$

$=\frac{x^4}{4}[logx-\frac{1}{4}]+c$.