Question

What is the integral of `[ln(lnx)]/[x] dx`?

Answer 1

I got:

`lnx*ln(lnx) - lnx + C`

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`f(x) = ln(lnx)/x`

Now we can do some unusual substitution...

First, recognize that `d(lnx) = 1/xdx`. Then, realize that `1/xdx` is in there. So, we have:

`f(u) = ln(u)du`
where `u = lnx` and `du = 1/xdx`.

Further considerations lead to using integration by parts on `int lnudu`. Let:

`s = lnu`
`ds = 1/udu`
`dt = du`
`t = u`

`intln(u)du`

`= st - int tds`

`= u*lnu - int u*1/udu`

`= u*lnu - u`

`= color(blue)(lnx*ln(lnx) - lnx + C)`