Density function of $XY$ when $X$ and $Y$ are uniform : What is wrong here? Let $X$ and $Y$ be independent ,uniform over [0,1].
I need to find the pdf of XY. So let $U = X, V= XY$ and Jacobian $J$ can be calculated as $J= 1/u$. So
$$f_{U,V}(u,v)  = f_{X,Y}(x,y)|J|= 1/u$$
So we get $f_V(v) = \int_0^1 \frac 1u du $ which does not exist.
Can anyone tell me where I am going wrong?
The answer in text book is different than what I am getting.

Your problem is that your integral is under the assumption that Y goes through 0 to $XY/X$ for a given X, which is not true because then Y would be allowed to be greater than 1 and in fact be unbounded. That's the reason why your integral blows up. You need to take care of this situation if you must go the Jacobian route.
Otherwise, as one comment states it's the easiest to get the answer with constructing CDF of XY as a double integral, and you'd do it through breaking the outer integral into two parts.
