Bound for the integral

Question

Is there any way to bound the following integral

$$\int_{-(\epsilon+1)/\sigma}^{(\epsilon-1)/\sigma} \mathrm e^{-t^2/2}\,dt$$

Answer 1

$$\int_{-(\epsilon+1)/\sigma}^{(\epsilon-1)/\sigma} \mathrm e^{-t^2/2}\,\mathrm dt\leqslant\sqrt{2\pi}\cdot(2\Phi(\epsilon/\sigma)-1)=\sqrt{2\pi}\,\mathrm{erf}(\epsilon/(\sigma\sqrt2))$$