I want to determine the values of $b>0$ such that $$\int_0^1\int_0^1(1-xy)^{-b}dydx$$ exists and is finite. I think that the integral is finite for $0<b<2$ and infinite for $b\geq 2$ but I am having trouble showing this. I was told we could bound the function from above and below if we consider $(1-(1-u)(1-v))$, where $x=1-u$ and $y=1-v$, but I have made no progress with this approach.
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For $n>-1$, we have $~\displaystyle\int_0^1\int_0^1\big(1-xy\big)^{n-1}~dy~dx=\dfrac{H_n}n,~$ where $H_n$ is the $n^{th}$ harmonic number. For $n\not\in$ N, please refer to this section. – Lucian Nov 10 '14 at 06:23