I need help with integral please

Question

Evaluate

Answer 1

integration by parts:

U = ln x dv = dx

dU = 1/x V = x

U V - integral( V du)

= x ln x - integral( x (1/x) dx)

= x ln x - integral( dx)

= x ln x - x

= x (ln x - 1)

check by differentiation:

x ( 1/x) + ln x - 1

1 + lnx - 1

ln x

limit as x--> e^2 : e^2 ( ln e^2 - 1) =

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

limit as x--> e : e ( ln e - 1) =

e ( 1 - 1) = 0

final answer: e^2 - 0 = e^2