integrable functions on their domain

dexter lab

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"Verify if both of this functions are integrable in their domains:
$g(x) = ( ||x||^2 + 1)^{ \frac{a}{b} }, \forall a>n$, domain in $ \mathbb{R} ^n$

$h(x) = ||x||^{-||x||}$, defined in $\mathbb{R}^3$"

I can't work this out. I believe I have to prove that there exists a majorant in the integral of the functions but I don't know how. Thank you.


IMPORTANT: I ask the admins to correct the notation because ai try LaTex notation but the preview post doesn't seem to be working. And ai don't know how to use it correctly. Thanks!"
 
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"Verify if both of this functions are integrable in their domains:
$g(x) = ( ||x||^2 + 1)^{ \frac{a}{b} }, \forall a>n$, domain in $ \mathbb{R} ^n$

$h(x) = ||x||^{-||x||}$, defined in $\mathbb{R}^3$"

I can't work this out. I believe I have to prove that there exists a majorant in the integral of the functions but I don't know how. Thank you.


IMPORTANT: I ask the admins to correct the notation because ai try LaTex notation but the preview post doesn't seem to be working. And ai don't know how to use it correctly. Thanks!"

First a matter of notation: Do you mean x to be a scalar for g(x)? If so, then I assume the the norm, ||x||, is really just the absolute value. Next, it appears also that you mean a and b to be integers and thus a/b is rational. For h(x) I would also assume the norm is the usual Euclidean norm. Assuming those are true it appears that your questions can be re-written as follows:

Are the following functions g and h integrable in closed form:
\(\displaystyle g(x) = ( x^2 + 1)^r\, \, \forall\, r\, \epsilon\, \mathbb{Q} \text{ and } x\, \epsilon\, \mathbb{R}\)

\(\displaystyle h(x) = ||x||^{-||x||}\, \, \forall\, x\, \epsilon\, \mathbb{R}^3\)



BTW: You need to put the LaTex code inside a [ tex ], [ /tex ] pair (without the spaces around the tex and /tex). If you do a Reply to this post with a "quote the post", you will see what I mean.
 
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