Does the following integral depending on $\alpha$ converge : $$\int_{R^2} \frac{1}{(1+x^4+y^4)^\alpha}$$ ?
I know that for $\alpha \le 0$ it diverges. It is pretty trivial since you can compare the function to $1$.
For $\alpha \ge 0$ I tried to change to polar coordinates and I got to $$\int_{0}^{2\pi} d\theta \int_{0}^{\infty} \frac{rdr}{(1+\frac{r^4}{2}(2-sin^2(2\theta))^\alpha}$$
I'm not pretty sure how to continue from here. I think I can bound it from both sides since $0 \le sin^2(2\theta) \le 1$ . I don't know exactly for which values of $\alpha$ I should use each bound.
Help would be appreciated.