reconciling intergration by parts with direct substitution

obviously one can use integration by parts to attempt to solve any integral not just those with funky products in the intergrand so with that in mind i was trying to solve the integral,

$\displaystyle

\int\frac{1}{xln(x)}dx

$

using integration by parts after i had already noticed that direct substituion of u=ln(x) would do the trick.

But with the following choices this is what I'm getting:

Let f(x) = $\displaystyle \frac{dx}{x}$, g(x) = $\displaystyle \frac{1}{ln(x)}$

Then

$\displaystyle

\int\frac{1}{xln(x)}dx $ = $\displaystyle \frac{ln(x)}{ln(x)} - \int\frac{ln(x)}{-xln(x)^2}dx$

= $\displaystyle 1 +\int\frac{1}{xln(x)}dx$

As you can see, this is nonsensical.

Would someone please tell me where I'm erring.

Thanks

The correct result is $\displaystyle

\int\frac{1}{xln(x)}dx = ln(ln(x)) + C $