Question

What is `int ln(lnx)/xdx`?

Answer 1

Consider the following substitution.

Let:
`t = lnx`
`dt = 1/xdx`

`intln(lnx)/(x)dx = intlntdt`

Now we can do an integration by parts.

`int udv = uv - intvdu`

Let:
`u = lnt`
`du = 1/tdt`
`dv = dt`
`v = t`

`=lnt - int t*1/tdt`

`= tlnt - t`

And now let's substitute back in our original variables.

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