I am using the fact that $X$ and $Y$ are independent if and only if $f_X(x)f_Y(y)=f(x,y)$. So I have
$f_{X} (x)=\int_{0}^{1}f(x,y)dy\\=\int_{0}^{1}x+ydy\\=[xy+\frac{y^{2}}{2}]_{y=0}^{y=1}\\=x+\frac{1}{2}$
and by basically exactly the same math, $f_Y(y)=y+\frac{1}{2}$. Then $f_X(x)f_y(y)=(x+\frac{1}{2})(y+\frac{1}{2})=xy+\frac{1}{2}(x+y)+\frac{1}{4}\ne f(x,y)$
And hence they are not independent. But can that be right? Why would the value of X have anything to do with the value of Y? It's not like one is a function of the other.
Or have I made a simple mistake? I looked through a couple of times and I'm pretty sure my math is right...