I am given the following joint pdf of the two random variables $X$ and $Y$: $$ f_{XY}(x,y) = \begin{cases} 1 & \text{for } 0 < x,y < 1 \\ 0 & otherwise \end{cases} $$ and let $U = X$ and $V=X/(1-Y)$.
Ultimately, I need to find the marginal densities $f_U(u)$ and $f_V(v)$.
First I solve for $X$ and $Y$ in terms of $U$ and $V$: $$ X = U \\ Y = -\frac{U-V}{V} $$
And I find that the support for $U,V$ is $0 \leq u \leq 1$, $0 \leq v \leq \infty$, $u \leq v$.
Now I find the joint pdf $f_{UV}(u,v)$: $$ J = \left|\begin{array}{cc} \frac{\partial x}{\partial u} & \frac{\partial y}{\partial u} \\ \frac{\partial x}{\partial v} & \frac{\partial y}{\partial v} \end{array}\right| \\ = \left|\begin{array}{cc} 1 & -\frac{1}{v} \\ 0 & \frac{u}{v^2} \end{array}\right| \\ = (1)(\frac{u}{v^2}) - (-\frac{1}{v})(0) \\ = \frac{u}{v^2} + \frac{1}{v} \\ |J| = \frac{u}{v^2} + \frac{1}{v} \\ $$
$$ f_{UV}(u,v) = f_{XY}\left(u, \ -\frac{u-v}{v}\right)|J| \\ = \begin{cases} 1\left(\frac{u}{v^2} + \frac{1}{v}\right) & \text{for support of } u,v \\ 0 & otherwise \end{cases} \\ = \begin{cases} \frac{1}{v} & 0 \leq u \leq 1, \ 0 \leq v \leq \infty, \ u \leq v \\ 0 & otherwise \end{cases} \\ $$
Now I attempt to find the marginal densities for $U$ and $V$: $$ f_V(v) = \int_{-\infty}^{\infty} f_{UV}(u,v) du \\ = \int_{0}^{1} \frac{u}{v^2} + \frac{1}{v} du \\ = \left[ \frac{u^2}{2v^2} + \frac{u}{v} \right]_0^{1} \\ = \left( \frac{1^2}{2v^2} + \frac{1}{v} \right) - \left(\frac{0^2}{2v^2} + \frac{0}{v} \right) \\ = \frac{1}{2v^2} + \frac{1}{v} \\ = \frac{1 + 2v}{2v^2} \\ $$
$$ f_U(u) = \int_{-\infty}^{\infty} f_{UV}(u,v) dv \\ = \int_{u}^{\infty} \frac{u}{v^2} + \frac{1}{v} dv \\ = \left[ -\frac{u}{v} + log(v) \right]_u^{\infty} \\ = \left( -\frac{u}{\infty} + log(\infty) \right) - \left( -\frac{u}{u} + log(u) \right) \\ = \left( -\frac{u}{\infty} + log(\infty) \right) - \left( -1 + log(u) \right) \\ = \infty \\ $$
But the integral for finding the marginal density of $U$ does not converge. Have I made a mistake in finding the support of $U$ and $V$, or in the process of finding the joint and marginal densities of $U$ and $V$?