How do you evaluate the integral #int lnxdx# from 1 to #oo#?

Question

1226 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2016-10-09T09:02:29

Integral does not converge

It seems pretty clear from the outset that this integral, which is the area under #y = ln x# from #1# to #oo#, is not going to converge

We can try to evaluate the integral using IBP

So

#lim_(t to oo) int_1^t lnx dx#

#= lim_(t to oo) int_1^t d/dx(x) lnx dx#

#= lim_(t to oo) ( [x lnx ]_1^oo - int_1^t x d/dx(ln x) dx)#

#= lim_(t to oo) ( [x lnx ]_1^oo - int_1^oo dx )#

#= lim_(t to oo) [x (lnx - 1) ]_1^oo = oo#