I am trying to get bound for the following integral $$ \int_0^{\infty}\frac{1}{|x|^r}dx, \mbox{for } 1\leq r< \infty $$ In particular, the bound of the form $\frac{constant}{r}$.
Sorry, we can omit the absolute value. And probably consider interval $(a,\infty)$ for some $a$ very close to 0. So, the integral is $$ \int_a^{\infty}\frac{1}{x^r}dx, \mbox{for } 1\leq r< \infty $$