Is there only numerical method to find this defenite integral or any other way?

definite-integrals

Here is an answer

$$\int_{a}^{b}e^{-x^2/2}dx= \sqrt {\frac{\pi}{2} }\left({{\rm erf}\left(\frac{b}{\sqrt{2}}\right)}-{{\rm erf}\left(\frac{a}{\sqrt{2}}\right)}\right),$$

where $\rm erf(x)$ is the error function

$$ \operatorname{erf}(x) = \frac{2}{\sqrt{\pi}}\int_{0}^x e^{-t^2}\,\mathrm dt. $$

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