Let $(X,Y)$ have joint density $f(x,y) = \frac{1}{2}(1+x+y)$ for $0<x<1$ and $0<y<1$.
So the joint density of $X$ and $U=X+Y$ is $f_{X,U}(x,u)=\frac{1}{2}(1+u)$. Now it is simple to get the domain of $X$ since it is provided in the problem description, but to get the domain of $U$ seems so be trickier since you need to extract the case of $U<1$ and $U\geq 1$. I need these values in order to get the marginal densities, but I am not quite sure what I need to do after I get $0<U-X<1$.