# Thread: why is this integral finite

1. ## why is this integral finite

Hello,

i want to show, that $\int_{\left||x\right||\leq 1} \left|P(x)\right|^{-q}*\left|x\right|^{-M}dx <\infty$

Here P is a polynomial of order m in $\mathbb{R}^n$ and M,q are some constants.
We know that $\int_{\left||x\right||\leq 1} \left|P(x)\right|^{-q} <\infty$ for some constant q.
But why is the first integral finite?
Regards

3. ## Re: why is this integral finite

I hope, you can see it now.

I have a idea, how to solve this problem. But i'm not sure, whether this way is correct.

We know that $\int_{\left||x\right||\leq 1} \left|P(x)\right|^{-q}*\left|x\right|^{-M}dx < \infty$
Can i conclude, that the set $S=\{x\in \mathbb{R}^n : |P(x)\right|^{-q}*\left|x\right|^{-M} has finite measure? Why is this correct?

Regards

4. ## Re: why is this integral finite

Take a close look at P(x). and q. Consider what you have and when it would be a problem.

If q >= 0, P(x) ^ q is just a polynomial (possibyl degenerate). That should be finite.
If q < 0, there would be a problem only if P(a) = 0 for some 'a' in [-1,1].
However, if it's going to be a problem for some 'a' in [-1,1], we have that last piece of information.

Did we get anywhere?