Find the marginal distribution of $V=X-Y$ Problem:
Show that the marginal density function of $f_V(v)$ if $V=X-Y $ is $$f_{V}(v)= \frac{1}{(1+|v|)^2}$$ for $ -\infty < v < \infty $.
When the bivariate density function $f_{X,Y}(x,y)$ is given as: 
$$f_{X,Y}(x,y) = \begin{cases}\frac{2}{(1+x+y)^3} \ &\text{ for } x,y>0\\0\ &\text{ otherwise } \end{cases}.$$
Attempt:
My solution strategy is quite clear as the joint density function for $f_{U,V}(u,v)$ can be obtained from the transformation formula for continuous variables as stated below:
$$f_{U,V}(u,v)=f_{X,Y}(x(u,v),y(u,v))\text{ }|\text{ }J \text{ }|$$
where $U$ is an auxiliary variable and $J$ the Jacobian. From this joint distribution the marginal density function $f_{V}(v)$ is derived by integration over the domain of the auxiliary variable $u$.
$$f_{V}(v)=\int_{\in u} f_{U,V}(u,v) \text{ }du $$
I am currently struggling with the choice of auxiliary variable $U$ and the integral that follows. So far my attempts include $U=Y$ and $U=X+Y$  with the corresponding density functions: $$f_{U,V}(u,v)=\begin{cases} \frac{2}{(1+v+2y)} \ & u = y\\ \frac{1}{(1+u)^3} \ & u=x+y\end{cases}$$
I am solving this as an exercise in my probability course. Any help is greatly appreciated! 

$$
f_{X,Y}(x,y) = \begin{cases}\frac{2}{(1+x+y)^3} \ &\text{ for } x,y>0\\0\ &\text{ otherwise } \end{cases}.
$$
What you want to compute is
$$
f_V(v)=\iint_0^\infty dx dy \frac{2}{(1+x+y)^3} \delta(v-(x-y))\ .
$$
Resolving the Dirac delta
$$
f_V(v)=\int_0^\infty dy \frac{2}{(1+(v+y)+y)^3}\Theta(v+y)\ ,
$$
where $\Theta(z)=1$ if $z>0$ and $0$ otherwise.
Therefore
$$
f_V(v)=\int_b^\infty dy \frac{2}{(1+(v+y)+y)^3}=\frac{1}{2 (2 b+v+1)^2}\ ,
$$
where $b=\max(0,-v)$.
So, for $v>0$
$$
f_V(v)=\frac{1}{2 (v+1)^2}
$$
and for $v\leq 0$
$$
f_V(v)=\frac{1}{2 (-v+1)^2}\ .
$$
We can check normalization
$$
\int_{-\infty}^\infty dv f_V(v)=\int_{-\infty}^0 dv\frac{1}{2 (-v+1)^2}+\int_0^\infty dv \frac{1}{2 (v+1)^2}=1\ ,
$$
as it should. [The pdf you wrote in your question for $V$ cannot be correct, as it is not normalized for $-\infty<v<\infty$]
