# Determining Independence of two random variables from joint density function

I know that to determine whether $$X$$ and $$Y$$ are independent I have to find the marginal distributions of $$X$$ and $$Y$$. I've already found the marginal probability density function $$f_Y$$, but I'm stuck on integrating the joint pdf with respect to $$y$$ to get the marginal probability density function $$f_X$$. I've already entered it into integral calculators and none of them were able to find the integral. Any help would be appreciated!

There is an easier way: $$f_Y(y)=Cye^{-y}\int_0^\infty e^{-x/y}dx=Cy^2e^{-y}$$ And, $$f_{X|Y=y}(x)=\frac{f_{X,Y}(x,y)}{f_Y(y)}=\frac{Cye^{-y-x/y}}{Cy^2e^{-y}}=\frac{1}{y}e^{-x/y}$$
The term $$y$$ hasn't vanished in the conditional density of $$X$$; so $$f_{X|Y=y}(x)\neq f_X(x)$$ in general, which means they're not independent.
There is an even easier way here. If $$X$$ and $$Y$$ are independent then the joint density kernel will be seperable, meaning that you can split it as:
$$f(x,y) \propto g(x) h(y).$$
In the present case you have a term $$e^{-x/y}$$ in the density, and there is no way to seperate this term into a product of functions of $$x$$ and $$y$$. Therefore your random variables are not independent.