$Q := (0,1) \times (0,1) \subset \mathbb{R}²$ and $f:Q \rightarrow\mathbb{R}$ with $f(x,y) = \frac{x-y}{(x+y)^3}$. Why $f$ isn't Lebesgue integrable?
A function is Lebesgue-Integrable if $\int \vert f \vert < \infty$. One option would be to compute this, if possible.
Is there an easier way to recognize it directly from $\vert f \vert$ ?.
Hint:
$$\int_{[0,1]^2} \left|\frac{x - y}{(x + y)^3} \right| > \int_0^1 \int_0^{\pi/2} \frac{r|\cos \theta - \sin \theta|}{r^3|\cos \theta + \sin \theta|^3}\, r\, dr\, d\theta \\= \int_0^1 \frac{dr}{r}\int_0^{\pi/2} \frac{|\cos \theta - \sin \theta|}{|\cos \theta + \sin \theta|^3}\, \, d\theta$$