Evaluating the improper integral $\int_0^1 \frac{\log (x \sqrt{x})}{\sqrt{x}} \,dx$ I am supposed to solve this integral but i have no idea how:

$$\int_0^1 \frac{\log (x \sqrt{x})}{\sqrt{x}} \,dx$$

Since one limit is $0$ it will be divided by zero.
Can someone please explain this to me (I really want to understand) and guide me through step by step. Thanks! :)

Use the transformation $x=z^2$, to get $$I=6\int_{0}^1 \ln z dz $$ and then use integration by parts to obtain $I=-6$.
