Question

find the integral of ln(lnx)/xlnx

Answer 1

Let u = ln(lnx).
derivative of lnu is u'/u, so ln(lnu) is the derivative of ln(u) divided by lnu, so that means
du = \[\frac{ \frac{ 1 }{ x } }{ lnx }=\frac{ 1 }{ xlnx }dx\]
Meaning dx = (du)(xlnx)

\[\frac{ u }{ xlnx }*du(xlnx)\]Leaving us finally with simply
\[\int\limits_{}^{}udu=\frac{ u^{2} }{ 2 }\]And since u = ln(lnx), the final answer is:
\[\frac{ (\ln(lnx))^{2} }{ 2 }+C\]