Question

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

Answer 1

Substitution Method.

Explanation:

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

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

Explanation:

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`