I=∫x(logx)2dx
Taking (logx)2 as first function and 1 as second function and integrating by part, we obtain
I=(logx)2∫xdx−∫[{(ddxlogx)2}∫xdx]dx
=x22(logx)2−[∫2logx⋅1x⋅x22dx]
=x22(logx)2−∫xlogxdx
Again integrating by parts, we obtain
I=x22(logx)2−[logx∫xdx−∫{(ddxlogx)∫xdx}dx]
=x22(logx)2−[x22−logx−∫1x⋅x2xdx]
=x22(logx)2−x22logx+12∫xdx
=x22(logx)2−x22logx+x24+C