How do you show whether the improper integral $int lim (lnx) / x dx$ converges or diverges from 1 to infinity?

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Answer 1 · 2016-06-06T20:04:00

$int_1^ooln(x)/xdx$ diverges to $oo$ .

$int_1^ooln(x)/xdx = lim_(N->oo)int_1^Nln(x)/xdx$

Let $u = ln(x) => du = 1/xdx$ .
Also, $x = 1 => u = 0$ and $x = N => u = ln(N)$ .

Making the substitution, we have

$lim_(N->oo)int_1^Nln(x)/xdx = lim_(N->oo)int_0^ln(N)udu$

$=lim_(N->oo)(u^2/2)_0^ln(N)$

$=lim_(N->oo)(ln^2(N)/2-0^2/2)$

$=lim_(N->oo)ln^2(N)/2$

$=oo$

Thus the integral $int_1^ooln(x)/xdx$ diverges to $oo$ .

How do you show whether the improper integral int lim (lnx) / x dxint lim (lnx) / x dx converges or diverges from 1 to infinity?