I am learning probabilities and I am not a mathematician. To calculate a marginal PDF I must integrate $1\over x$ (a conditional uniform PDF) over the range $(0,1]$. The integral of $1\over x$ is $\ln x$ and $\ln x$ at $0$ takes an infinite value. So I am stuck.
Your advice will be appreciated.
Note: Indeed, I ended up calculating the wrong interval. The original problem was that a random variable X is uniformly distributed over [0, y] and Y is uniformly distributed over [0, 1]. So what is the marginal of X?
Using the total probability theorem I integrated the marginal of Y (1) multiplied by the conditional of X given Y (1/y). However I set as limits for the definite integral the range [0, 1] when in reality it is [x , 1] since Y can not be less than X for any given X.
The integral doesn't converge because $$\lim_{\epsilon \downarrow 0} \log(\epsilon) = - \infty$$