∫x ln(x)dx

Let: v = x 2/2, dv = x*dx.

Let u=ln(x),

Then ∫xln(x)dx

=∫ln(x)*d(x^2/2)

=(x^2/2)ln(x)-∫(x^2/2)d(ln(x))

=(x^2/2)ln(x)- 1/2∫xdx

=(x^2/2)ln(x)-x^2/4+C

Namely:

∫xln(x)dx =(x^2/2)ln(x)-x^2/4+c①

Here c is an arbitrary integer constant.

For the definite integral of ∫xlnx dx with the upper limit of e and the lower limit of 1,

Formula ① can be used for calculation.

∫( 1→e) x lnx dx

=(x^2/2)ln(x)|( 1→e)- x^2/4|( 1→e)

= (e^2/2) - ( 1/4 - e^2/4)

= e^2/4 - 1/4