Exercise 9, § 2

Let $\alpha$ be a number $> 0$ and $f$ a function defined on the interval $]0, +\infty[$, $\geq 0$ and decreasing, and such that the integral $\int_0^{+\infty} t^\alpha f(t) \, dt$ converges. Show that for all $x > 0$ one has

(First prove this when $f$ is constant on an interval $]0, a]$ and zero for $x > a$, then for a sum of such functions, and pass to the limit for the general case.)