I has just been introduced the definition of "improper integral" and proved that: $$\int\_{1}^{+\infty} x^{r}dx$$
converges if $r<-1$.

Then, why is it not so for $0$ to $+\infty$? Because we have
$$ \int\_{0}^{x} t^{r}dt = \int\_{0}^{1} t^{r}dt + \int\_{1}^{x} t^{r}dt. $$
Therefore: $$\int\_{0}^{+\infty} x^{r}dx = \lim\_{x\rightarrow +\infty} \int\_{0}^{x} t^{r}dt = \int\_{0}^{1} t^{r}dt + \lim\_{x\rightarrow +\infty}\int\_{1}^{x} t^{r}dt$$ exists. So $\int\_{0}^{+\infty} x^{r}dx$ converges??