Why $f(x)$ disappears in the probability exercise bellow?

Question

In the image bellow, in the solution part, why does $f(x)$ disappears from the first line to the second?

enter image description here

Answer 1

It is because $X$ is a uniform-$(0,1)$ random variable. Its pdf, $f_X$, is defined by

$$f_X(x) = \begin{cases} 1 & x \in (0,1) \\ 0 & \text{otherwise} \end{cases}$$

It's also why you see the integral bounds change from $(-\infty,\infty)$ to $(0,1)$.