Question

the area bounded by y = 1nx and the lines y = 0 and x=e

Answer 1

Did you mean

y = ln x ? , I will assume you did

Make a sketch to see the region in question.
so you want
∫ lnx dx from x = 1 to e
= [xlnx - x] from 1 to e
= elne - e - (1ln1 - 1)
= e - e - 0 + 1
= 1

(the integral of lnx should be in your repertoire of standard integrals)