$X$ - random variable, $P(X<1)=0$ and $P(X>e)=0$, $y \in [1,e]$ and $P(X<y) = \int\limits_1^y \frac{1}{x} \ dx$.
Is it correct to find $E[X]$ in the following way?
$ F_X(y) = \int\limits_1^y \frac{1}{x} dx = \ln(y) $
$ f_X(y) = \frac{1}{y} $
$ E[X] = \int\limits_1^e f_X(y)y \ dy = \int_1^e dy = e - 1$