How can I integrate the density function g(x,y) over positive reals in R^2?

[jam_33]

Homework Statement

f(x) is a density on R+ so f(x) < 0 if x < 0. Define g_(X,Y)(x,y) = f(x+y)/(x+y). Show g is a density on R^2.

Homework Equations

the first part is easy (showing that g is in fact >= 0. The part I am struggling with is the double integral of the g(x,y) over the positive reals.

The Attempt at a Solution

I have tried substitution as well as by parts but I always end up with something I can't integrate. Any suggestions on how to attach the problem?I'm guessing I need to give more information to get a response. I am struggling getting going on the actual problem...I believe the substition i used (u = x+y) is wrong as I get a very nasty integral. I am just looking for some advice on how to get going

 

[Billy Bob]

Welcome to PF, jam_33.

I would try converting to polar coordinates.

 

[jam_33]

Welcome to PF, jam_33.

I would try converting to polar coordinates.

Hello Billy Bob,

Thanks for the suggestion as it worked!

cheers,

jam_33