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> Determine for which real numbers $a\_1,a\_2,\dots,a\_n$ the following integral is finite:
>  $$\int\_{\mathbb{R}^n} \frac{1}{1+|x\_1|^{a\_1}+|x\_2|^{a\_2}+\dots+|x\_n|^{a\_n}}dx. $$ 
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I was stuck on this homework problem and looking for some help. 
I tried to use the change of variable by defining $G(x\_1,x\_2,\dots,x\_n) = y\_i $ for $ |x\_i|^{a\_i}$. then the integral $$\int\_{\mathbb{R}\_+^n} = \int\_{G^{-1}(\mathbb{R}\_+^n)}f(G\_n(x))|det\,
 D\_xG|dx \\= \frac{1}{a\_1a\_2\dots a\_n}\int\_{\mathbb{R}^n\_+}\frac{1}{1+x\_1+x\_2+\dots+x\_n} x\_1^{\frac{1}{a\_1}-1}x\_2^{\frac{1}{a\_2}-1}\dots 
x\_n^{\frac{1}{a\_n}-1}dx$$
and then I don't know how to proceed. I feel like this way is an dead end.