Integral por parte
=-∫lnxd[1/(x-1)]
=-lnx/(x-1)+∫1/ [x(x-1)]dx
=-lnx/(x-1)+∫[1/(x-1)-1/x]dx
=- lnx/(x-1)+ln|x-1|-ln|x|(0,e^2)
Cuando x→0
lim-lnx/(x -1)+ln|x-1|-ln|x|
=lim-lnx/(x-1)+0-lnx
=lim(-lnx-xlnx +lnx)/(x-1)
=lim-xlnx/(x-1)
Y para f(x)=xlnx x→0
=lnx/(1/x), pertenece a ∞/∞
Usa la regla de Robita
=lim(1/x)/(-1/x^2) x → 0
=lim-x x→0
=0
Entonces lim-lnx/(x-1)+ln|x-1|- ln |x|
Cuando x=e^2,
-lnx/(x-1)+ln|x-1|-ln|x=e^ 2
=-2/(e^2-1)+ln(e^2-1)-2
=ln(e^2-1)-2e^2 /( e^2-1)
Entonces, la fórmula original =ln(e^2-1)-2e^2/(e^2-1)