Improper Multivariable Integrals How can I find the values of $\alpha$ for which the following integrals (in $\mathbb{R}^n $ ) converge ? 


*

*$\int_{|\vec{x}|\geq 1 } \frac{ln(|\vec{x}|^3 )}{|\vec{x}|^\alpha} d\vec{x} $ 

*$\int_{\mathbb{R}^n } \frac{sin(|\vec{x}|)}{|\vec{x}|^\alpha} d\vec{x} $ 
I guess the thing is we need to calculate the limit form of these integrals, in some other coordinate system, where $r= ||\vec{x}|| $ ... I can't figure out what will be that Jacobian in this case and can't understand how to solve these questions
Will you please help me?
Thanks in advance
 A: Recall that $\def\vol{{\rm vol}}\vol_{n-1}(rS^{n-1}) = 2\frac{\pi^{n/2}}{\Gamma(n/2)} r^{n-1} =: \beta_{n-1} r^{n-1}$, we have $\def\abs#1{\left|#1\right|}$ \begin{align*}
  \int_{\abs x \ge 1} \frac{\log \abs x^3}{\abs x^\alpha}\, dx 
  &= \int_{1}^\infty \int_{rS^{n-1}} \frac{\log r^3}{r^\alpha}\, dS(x)\, dr\\
  &= \beta_{n-1} \int_1^\infty \frac{\log r^3}{r^\alpha} r^{n-1}\, dr\\
  &= 3\beta_{n-1}\int_1^\infty {r^{ n - 1-\alpha}\cdot \log r}\, dr
\end{align*}
This converges if $n-1-\alpha < -1$, that is $\alpha > n$.
For the second case, arguing along the same lines, we have
\begin{align*}
  \int_{\mathbb R^n} \frac{\sin\abs x}{\abs x^\alpha}\, dx
  &= \int_0^\infty \frac{\sin r}{r^\alpha}\beta_{n-1} r^{n-1}\, dr\\
  &= \int_0^\infty \sin r \cdot r^{n-1-\alpha}\, dr
\end{align*}
This converges "at $\infty$" if $n-1-\alpha < -1$, that is $\alpha > n$, and "at 0" if $n-\alpha> -1$ (note that $\sin r\cdot r^{n-1-\alpha} = \frac{\sin r}r \cdot r^{n-\alpha}$, that is if $\alpha < n+1$. So we must have $\alpha \in (n, n+1)$. 
