You started off decent enough.
\(\displaystyle \int \frac{\sqrt{u}}{u-1}du\)
Let \(\displaystyle t=\sqrt{u}, \;\ t^{2}=u, \;\ 2tdt=du\)
This gives:
\(\displaystyle \int\frac{2t^{2}}{t^{2}-1}dt\)
Now, use partial fractions and expand:
\(\displaystyle 2\int dt+\int\frac{1}{t-1}dt-\int\frac{1}{t+1}dt\)
Integrate and back sub.
