# How to prove that this integral converges absolutely?

$f:[a,{\infty}[\to\mathbb{R}\$ is bounded and suppose that $f$ is integrable on each interval of the form $[a,b[$. Prove that $$\int_0^\infty \frac { f(x) }{ x^p } \ \, dx$$ converges absolutely for $p>1$. Conclude that, if $x^pf(x)$is bounded for some $p>1$, then $$\int_a^\infty f\ \, dx$$ converges absolutely.

• Did you mean $[a,b]$ – Justin Oct 13 '13 at 2:03
• That might not be integrable. Consider $f(x)=1$. – Jack M Oct 13 '13 at 2:24