Two-sided inequality for e^(x^2) integral from x to infinity e^(-t^2) dt for non-negative real x

Theorem

The following formula holds for $x\geq 0$: $$\dfrac{1}{x+\sqrt{x^2+2}}< e^{x^2} \displaystyle\int_x^{\infty} e^{-t^2} \mathrm{d}t \leq \dfrac{1}{x+\sqrt{x^2+\frac{4}{\pi}}},$$ where $e^{x^2}$ denotes the exponential, and $\pi$ denotes pi.

References

• 1964: Milton Abramowitz and Irene A. Stegun: Handbook of mathematical functions ... (previous) ... (next): 7.1.13