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Show that $f:\Bbb R^n\to\Bbb R$, given by: $$ f(x) = \begin{cases}\sin\left(\frac{1}{\|x\|}\right)\|x\|^{-n-\arctan(\|x\|-1)} & x\not=0 \\ 0 & x=0 \\ \end{cases}$$ is Lebesgue integrable.

I don't know how to prove this, I want to use a corolary of the dominated convergence theorem, but I just got really confused. I considered the family of rings given by $A_k:=\{x\in\Bbb R^n:\frac{1}{k}\le \|x\|\le k\}$, then $A_k\subset A_{k+1}$ and $\cup_{k\in\Bbb N} A_k=\Bbb R^n-\{0\}$. Now if we consider the function $$f|_{A_k}(x)=\begin{cases} \sin\left(\frac{1}{\|x\|}\right)\|x\|^{-n-\arctan(\|x\|-1)} & \|x\|\in[1/k,k] \\ 0 & \|x\|\not\in[1/k,k] \end{cases}$$ then is well defined and we want it to be Lebesgue-integrable in $A_k$ for any $k\in\Bbb N$, however I don't know how to do this.

There's also this other exercise, that says:

Let $X_1,X_2$ be integrable in $\Bbb R^n$ such that $X_1\cap X_2$ is null. If $X=X_1\cup X_2$ then $f:X\to\Bbb R\cup\{\pm\infty\}$ is integrable in $X$ $\iff$ $f|_{X_i}$ is integrable in $X_i$ for $i=1,2$, and $\int_X f=\int_{X_1}f+\int_{X_2}f$

I think that is almost the same problem, we have to show that the restricted function is integrable. I'm really lost with this problem...

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