Let k, s be natural numbers, prove that the function :
f(x) = x^(-k) * sin(x^s) for x in (0,1] and f(x) = 0 for x=0
is integrable in [0,1] if and only if k<=s.
Thank you!
Note that f is continuous and positive on (0,1).
If $\displaystyle s \ge k$, then we have
$\displaystyle \int_0^1\frac{\sin(x^s)}{x^k}dx < \int_0^1\frac{x^s}{x^k} \, dx = \int_0^1 x^{s-k}dx < \infty .$
If $\displaystyle s < k$, note that
$\displaystyle \lim_{x \to 0^{+}}\frac{\sin(x^s)}{x^k}=\infty.$