I can see that

$\displaystyle \int\frac{1}{u} du$

is easier to integrate, and can carry that integration out.

However, I can't see how your:

$\displaystyle \int\frac{\frac{1}{3}(u-2)}{u}\frac{1}{3} du$

corresponds to

$\displaystyle \int\frac{1}{u} du$

or how you got there.