I am learning basic calculus, and have learnt how to substitute terms in integrals to make it easier to work with (let $u=f(x)$, then $dx=\frac{du}{f'(x)}$). During classes, our lecturer presented something like the following steps repeatedly without explaining how it works, and my not-so-bright brain just can't figure out what is happening. Here's a basic example:
$$\int \sqrt x \ln x dx$$ $$=\int \ln x d(\frac{2}{3}x^{\frac{3}{2}})$$
I figured out that $\frac{d}{dx} \frac{2}{3}x^{\frac{3}{2}} = \sqrt x$, but why does this work? What am I substituting (what is $f(x)$ substituted as $u$)? When it's used in a definite integral, how do I change the two bounds?
\begin{align} \int\sqrt{x}\ln{(x)}\mathrm{d}x &=\int\ln{(x)}\mathrm{d}u\qquad(u=\frac23x^{3/2},\quad\mathrm{d}u=\sqrt{x}\mathrm{d}x)\\ &=\int\frac23\ln{\left(\frac32u\right)}\mathrm{d}u\\ \end{align}