Suppose \(X\) and \(Y\) are jointly absolutely continuous random variables with joint density

\[ f_{XY}(x,\,y)=xe^{-x(y+1)},\quad x,\,y>0. \]

1. Find the marginal density of \(X\).
2. Find the marginal density of \(Y\).
3. Are \(X\) and \(Y\) independent?
4. Find the conditional density of \(X\) given \(Y=y\).
5. Find the conditional density of \(Y\) given \(X=x\).