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! :)

There are two things I would do first looking at this integral.
The first is resolve the $x\sqrt{x}$ into $x^\frac{3}{2}$, then use the log laws to write $\frac{3}{2}\log(x)$.
This gives you a much nicer log expression, which then makes integration by parts a much easier process.
