<s>[INST] How to apply Fubini's theorem? I was asked to show the equality of these integrals 
$$\int_0^1\int_0^1\frac{x^2-y^2}{(x^2+y^2)^{3/2}}\log(4+\sin x)dydx
  =\int_0^1\int_0^1\frac{x^2-y^2}{(x^2+y^2)^{3/2}}\log(4+\sin x)dxdy\tag{1}$$
Which can be answered by using Fubini's theorem but in order to use Fubini here is what I did
Let $f(x,y)=\frac{x^2-y^2}{(x^2+y^2)^{3/2}}\log(4+\sin x)$. Now 
\begin{align*}
|f(x,y)|&= \biggl|\frac{x^2-y^2}{(x^2+y^2)^{3/2}}\log(4+\sin x)\biggl|\\
        &\leq \log5\biggl|\frac{x^2-y^2}{(x^2+y^2)^{3/2}}\biggl|\\
        &\leq\log 5\biggl(\frac{x^2+y^2}{(x^2+y^2)^{3/2}}\biggl)=\frac{\log5}{(x^2+y^2)^{1/2}}:=g(x,y)
\end{align*}
To apply Tonelli Theorem on a non negative function 
$$\int_0^1\int_0^1\frac{\log5}{(x^2+y^2)^{1/2}}dxdy\tag{2}$$
This is the part I get stuck. If the above integral in $(2)$ is finite, then we can apply Fubini and $(1)$ follows immediately.  Is my choice of $g(x,y)$ fine? If so, how can I evaluate $(2)$. But if not, what is the best choice to bound $|f(x,y)|$?
 [/INST] Hint: change to polar coordinates.
</s>