#int_1^2 x^3lnabsx \ dx#
#= int_1^2 x^3 ln x \ dx#, as #x >0#
#= int_1^2 d/dx(x^4/4) ln x \ dx#
#= [ x^4/4 ln x ]_1^2 - int_1^2 x^4/4 d/dx( ln x )\ dx#
#= [ x^4/4 ln x ]_1^2 - int_1^2 x^3/4 \ dx#
#= [ x^4/4 ln x - x^4/16 ]_1^2#
#= [ 16/4 ln 2 - 16/16 ] - [ 1/4 (0) - 1/16 ]#
#= 4 ln 2 - 15/16 #