I haven't found a satisfactory explanation of the meaning of $P(X\in dx)$.
I believe it means $P(x\le X \le x+dx)=\int _x ^{x+dx}f(t)dt$ where $f$ is the density. However, the meaning of $dx$ is ambiguous to me, and I don't see how to use this to computute a pdf.
Can I obtain a precice definition of this expression and see how it is used to compute the PDF of any well-known distribution?