Quick question on marginal density function

Question

Question is:

$f(x,y)=e^{(-x/y)}e^{-y}/y$ when $x>0, y>0$
it equals to 0, otherwise

What is $f_{X}(x)?$

I think: $f_{X}(x)= \int^{∞}_0f(x,y)dy = \int^{∞}_0 e^{(-x/y)}e^{-y}/y dy$

Then I should consider y as a variable,

x as constant.

but it seems I can't separate x, y. Since the term is

$ e^{[-(x+y^2)/y]} /y dy$

if we consider

$ (e^z)' = e^z $ seems not working easily here,

since we have

$z = [-(x+y^2)/y]$

Then it still needs to consider $z'$ into this equation...

I really really stuck on here.

Is there anyone could help me?

Many thanks!