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

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

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

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

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