I'm trying to determine wethere the following integral converges:
$$\iint _{\Bbb R^2} e^{-(x+y)^4}dxdy$$
I tried to substitute $u=x+y$, $v=x$ (is this diffeomorphism?) and got $$\iint _{\Bbb R^2} e^{-u^4}dudv$$ and this diverges. But according to wolframalpha, it should converge. What did I do wrong?
$$\iint _{\Bbb R^2} e^{-(x+y)^4}dxdy=\iint _{\Bbb R^2} e^{-r^4(\sin\theta+\cos\theta)^4}rdrd\theta=\iint _{\Bbb R^2} re^{-4r^4\sin^4(\theta+\frac{\pi}{4})}drd\theta=\iint _{\Bbb R^2} re^{-4r^4\sin^4\theta}drd\theta=\iint _{\Bbb R^2} e^{-4y^4}dxdy=\int_{\Bbb R}(\int_{\Bbb R}e^{-4y^4}dy)dx=\int_{\Bbb R}Kdx$$where $K$ is a positive real number so the integral doesn't converge at all. It is also clear from the sketch below: