Question

Integrate the function: $x^2 log\ x$

Answer 1

Let $I$ =∫$$x^2 log\ x\ dx

Taking $log\ x$ as first function and $x^2$ as second function and integrating by parts, we obtain

$I= log\ x∫[x^2dx-∫{(\frac {d}{dx}log\ x)∫x^2dx]}dx$

$I= log\ x(\frac {x^3}{3})-∫\frac 1x.\frac {x^3}{3}dx$

$I=\frac { x^3log\ x}{3}-∫\frac {x^2}{3}dx$

$I= \frac {x^3log\ x}{3}-\frac {x^3}{9}+C$