Let us consider the random vector $(X,Y)$ with pdf given by $$f(x,y)=\frac{5}{2}e^{-x-2y} \quad for \quad 0<x<+\infty, 0<y<2x.$$ Find the pdf of $X$ and the pdf of $Y$.
I calculate the integrals
$$f_X(x)=\int_{0}^{2x}\frac{5}{2}e^{-x-2y}dy=\frac{5}{4}e^{-x}-\frac{5}{4}e^{-5x} \quad \text{for} \quad x \in (0,+\infty)$$ and $$f_Y(y)=\int_{\frac{y}{2}}^{+\infty}\frac{5}{2}e^{-x-2y}dx=\frac{5}{2}e^{-\frac{5}{2}y} \quad \text{for} \quad y \in (0,+\infty).$$
But in the textbook the solution is different, namely $f_Y(y)=\frac{5}{2}e^{-\frac{3}{2}y}$ for $x \in (0,+\infty)$. Am I doing something wrong?
The answer given is wrong and it is not even a density function!. Your answer is right.