Can you help with this integration problem? Integration of $\displaystyle \int \dfrac{x}{x^2-\sqrt{x}} dx.$
I attempted substituting $\sqrt{x}$ with $u$ and then solving it using partial fractions but I'm not getting the right answer. I know I could used conjugates but I want to understand why the way I solved it at first was wrong. 

You were by the right way, let me continue your work, if 
$$I=\displaystyle \int \dfrac{x}{x^2-\sqrt{x}} dx.$$
First you make $u=\sqrt{x}$, then $du=\dfrac{1}{2\sqrt{x}}dx$, so
$$I=2\displaystyle \int \dfrac{u^2}{u^3-1} du.$$
Then, you make the substitution $s=u^3-1$, so $ds=3u^2du$ and
$$I=\frac{2}{3}\displaystyle\int \dfrac{1}{s}ds = \frac{2}{3} \ln(s)+C $$
From that
$$I=\frac{2}{3} \ln(u^3-1)=\frac{2}{3} \ln(\sqrt{x}^3-1)+C$$
