I would like to prove some bound of the form
$$\left|I_k(x)\right| := \left|\int_0^x t^k \sin{t}dt\right| \leqslant Cx^k$$
for a constant $C$ independent of $k$ and $x$. I think $C=1$ should be true whenever $k > 2$ however I am unable to prove it.
It might be useful to remark that following recurrence relation holds
$$I_{k+2} (x) = -x^{k+2} \cos x + (k+2)x^{k+1}\sin x - (k+1)(k+2) I_k(x)$$
and we have the special values
$$I_0(x) = 1-\cos x$$
$$I_1(x) = \sin x - x \cos x$$
Many thanks