The mistake is in there : |J(x,y)|=|x|/y²
The absolute value will change the boundaries.
I trust you for the formula of the change of variables. I've never seen it this way, so I can't really check.
If X and Y are independent standard random variables, determine the joint density function of
Then use your result to show that is a Cauchy distribution.
However when you evaluate the integral the answer seems to be meaningless