For which values of $a\in\mathbb{R}$ the following integral converges?
$\iint_D{\frac{1}{(\sqrt{1-(x^2+y^2)})^{5a}}}dxdy$
$D=\{(x,y):\sqrt{x^2+y^2}<1\}$
My attempt:
Changing to polar coordinates we have:
$2\pi\int_{0}^{1-\frac{1}{n}}\frac{r}{(1-r^2)^{5a/2}}dr$
And then I don't know how to solve it... any suggestion would be nice.
My english isn't very good; if you find any errors, please correct them.
Thanks.
That's a good start. You can evaluate the integral over $r$ using a $u$-substitution. If $u=1-r^2$, then $du=-2rdr$ and the integral becomes $$\lim_{n\to 0+}-\pi \int_1^n\frac{du}{u^{5a/2}}=\lim_{n\to 0^+}\pi \int_n^1\frac{du}{u^{5a/2}}$$ For which values of $a$ does this integral converge?