I wonder if anyone can show me a neat way to do integrate out x and then y to obtain marginal pdfs given the joint

 f_{X,Y} (x, y) = y \cdot \exp(-xy - y)


 x \geq 0, y \geq 0

I'm getting tied in knots with integrating by parts again and again. I think there must be a simpler way.

Thanks for any help.