Let's start with the direction f integrable implies f\^3 integrable.
Say f is defined on \[a,b\] and takes values in \[-A,A\]. For a given partition of \[a,b\], the difference between the upper and lower sums of f will be defined in terms of the numbers M\_i - m\_i, where M\_i and m\_i are respectively the sup and inf of f on the i-th subinterval.
You want to show that this difference can be made arbitrarily small for f\^3 provide this can be done for f. This will be accomplished if you can show that |M\_i\^3 - m\_i\^3| <= K|M\_i - m\_i| for some constant K. But this is true because the function g(x) = x\^3 is C\^1 on \[-A,A\].
For the converse, things are a little more complicated because the derivative of h(x) = x\^(1/3) is infinite at 0. However, it's still possible to prove that for any fixed eta > 0, there is a constant K, depending on eta, such that |h(x) - h(y)| <= eta + K|x - y| for x, y in \[-A\^3,A\^3\]. Now if you want the difference between the upper and lower sums for f to be less than a prescribed epsilon > 0, first you can pick eta small enough that eta.(alpha(b) - alpha(a)) < epsilon/2, and then select a partition of \[a,b\] such that the difference between the upper and lower sums for f\^3 is < epsilon/2K.