# Math Help - Integrals problem

1. ## Integrals problem

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!

2. Note that f is continuous and positive on (0,1).

If $s \ge k$, then we have

$\int_0^1\frac{\sin(x^s)}{x^k}dx < \int_0^1\frac{x^s}{x^k} \, dx = \int_0^1 x^{s-k}dx < \infty .$

If $s < k$, note that

$\lim_{x \to 0^{+}}\frac{\sin(x^s)}{x^k}=\infty.$