I would like to calculate $$\int\_{0}^{\infty} \frac{ x^2 \log(x) }{1 + x^4}$$ by means of the Residue Theorem. This is what I tried so far: We can define a path $\alpha$ that consists of half a half-circle part ($\alpha\_r$) and a path connecting the first and last point of that half circle (with radius $r$) so that we have $$ \int\_{-r}^{r} f(x) dx + \int\_{\alpha\_r} f(z) dz = \int\_{\alpha} f(z) dz = 2 \pi i \sum\_{v = 1}^{k} \text{Res}(f;a\_v) $$ where $a\_v$ are zeros of the function $\frac{x^2 \log(x) }{1+x^4}$. 

If we know $$\lim\_{r \to \infty} \int\_{\alpha\_r} f(z) dz = 0 \tag{\*} $$ then we know that $$\lim\_{r \to \infty} \int\_{-r}^{r} f(x) dx = \int\_{-\infty}^{\infty} f(x) dx = 2 \pi i \sum\_{v=1}^{k} \text{Res}(f;a\_v) $$ and it becomes 'easy'. 

**Q**: How do we know (\*) is true?