If f is integrable on $[a,b]$, prove $f^{q}$ is integrable on $[a,b]$ Let $q $ be a rational number. Suppose that $a < b,\ 0 < c < d $, and that $f : [a,b] $->$ [c,d] $. If $f$ is integrable on $[a,b]$, then prove that $f^{q}$ is integrable on $[a,b]$. 
I think that the proof involves the binomial theorem. My book has a proof showing that $f^2$ is integrable if $f$ is integrable, which I assume can be easily extended to $f^n$ for any integer $n$. I'm not sure exactly how to go about it from there though. 

Hints:

*

*$f$ is Riemann integrable and $g$ is continuous then the composition $g \circ f$ -- when this makes sense --is Riemann integrable (Theorem 8.18 of these notes).


*If $f: [a,b] \rightarrow [c,d]$ is continuous and monotone, then the inverse function $f^{-1}$ exists and is continuous (Theorem 5.39 of loc. cit.).
Alternately, as with so many of these kinds of results, this is an immediate corollary of Lebesgue's Criterion for Riemann Integrability (Theorem 8.28 of loc. cit.).  In fact this latter route takes care of irrational exponents as well.
