Express $\int_{\sin nx}^{\sin(n+1)x}\sin t^2dt$ in terms of $x$ and $n$

Question

Please help me to express $$\int_{\sin nx}^{\sin(n+1)x}\sin t^2\,dt$$ in terms of $x$ and $n$. If it is not possible please help to establish bounds on the integral again in terms of $x$ and $n$. The reason for question is another (more complex) question involving uniform convergence of a series I asked here.

Answer 1

By substitution $t^2=\frac12\pi u^2$ we get

$$\int \sin(t^2)\,dt=\sqrt\frac\pi2\int\sin\left(\frac12\pi u^2\right)\,du=\sqrt\frac\pi2 S(u) = \sqrt\frac\pi2 S\left(\sqrt\frac\pi2 t\right)$$

So

$$\int_{\sin(nx)}^{\sin((n+1)x)} \sin(t^2)\,dt= \sqrt\frac\pi2 S\left(\sqrt\frac\pi2 \sin((n+1)x)\right)- \sqrt\frac\pi2 S\left(\sqrt\frac\pi2 \sin(nx)\right)$$