Expectation of X squared over Y for a product density

The random pair $(X,Y)$ has joint density $f(x,y)=10\,y\,e^{-5x}\quad\text{for } x>0,\ 0<y<1,$ and zero elsewhere. Find $\mathbb{E}\!\left[X^2/Y\right]$.

$\tfrac4{25}$. The density factors, so $X\sim\text{Exp}(5)$ and $Y$ has density $2y$ on $(0,1)$, independently. Then $\mathbb{E}[X^2]=\tfrac{2}{25}$ and $\mathbb{E}[1/Y]=\int_0^1 2\,dy=2$, and the product is $\tfrac4{25}$.