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

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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

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

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

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

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

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

How do you evaluate the integral int lnxdxint lnxdx from 1 to oooo?