Hello, I'd like some help on this tough problem. Thanks in advance!

Given the joint density:

find the joint density function

and find out if U, V are independent.

My work for this part of the problem:

I got a joint density function

is that correct? and also, how do I integrate that with the crazy exponents? or is there a simpler way to test for independence other that checking

?

The second part of the problem:

Let the transformation from (X,Y) -> (R,θ) be defined as

find the joint density and marginal densities.

So far on this part I have it solved for X and Y:

I tried plugging in the equations into each other but I don't know how to get rid of the tan(). Any help is appreciated.