Find $int \ ln(lnx)/x \ dx$ ?

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Answer 1 · 2017-12-04T13:30:34

Substitution Method.

Let $log x=t$
Now differentiate this on both sides.
$implies$ $1/x dx=dt$
Now, $int logx$ is done by product rule.
Write $logx$ as $1*logx$

Answer 2 · 2017-12-04T14:08:23

$int \ ln(lnx)/x \ dx = lnxln(lnx)-lnx + c$

Assuming logarithm base $e$ , We seek:

$I = int \ ln(lnx)/x \ dx$

We can perform a substitution:

$u= lnx => (du)/dx=1/x$

Substituting into the integral we get:

$I = int \ lnu \ du$

This is now a standard integral (and can be readily derived with an application of Integration By Parts), and we have:

$I = u lnu-u + c$

And restoring the substitution we have:

$I = lnxln(lnx)-lnx + c$

Find int \ ln(lnx)/x \ dx int \ ln(lnx)/x \ dx?