Question

Integrate the function: \(x(logx)^2\)

Answer 1

The correct answer is: \(\frac{x^2}{2}(logx)^2-\frac{x^2}{2}logx+\frac{x^2}{4}+C\)
Let \(I=∫x(logx)^2 dx\)
Taking\((logx)^2\) as first function and 1 as second function and integrating by parts,we
obtain
\(I=(logx)^2∫x.dx-∫[{\frac{d}{dx}logx)^2}∫x dx]dx\)
\(=\frac{x^2}{2}(logx)^2-[∫2logx.\frac{1}{x}.\frac{x^2}{2}.dx]\)
\(=\frac{x^2}{2}(logx)^2-∫xlogx. dx\)
Again integrating by parts,we obtain
\(I=\frac{x^2}{2}(logx)^2-[logx∫x dx-∫[{(\frac{d}{dx}logx)∫x dx}]dx]\)
\(=\frac{x^2}{2}(logx)^2-[\frac{x^2}{2}-logx-∫\frac{1}{x}.\frac{x^2}{2}dx]\)
\(=\frac{x^2}{2}(logx)^2-\frac{x^2}{2}logx+\frac{1}{2}∫x dx\)
\(=\frac{x^2}{2}(logx)^2-\frac{x^2}{2}logx+\frac{x^2}{4}+C\)