I need to prove the following inequality: $$\int_0^x \sin(t^{\alpha})dt>0$$ for $x>0$ and $\alpha>1$.
I used Mathematica to plot the integrals for some values of $\alpha=2,4/3,3$ ($\alpha=2$ is the well-known Fresnel integral). It looks like the claim it correct. But I don't know how to prove it. Does anyone have any suggestions or ideas?