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

Is there only numerical method to find this integral or any other way? Any references or links would be helpful.

$$\int _{a} ^ {b} e ^ {-x^2/2} dx .$$

• $a$ and $b$ are constants – dexterdev Sep 24 '13 at 15:00
• You can get an answer in terms of error function. – Mhenni Benghorbal Sep 24 '13 at 15:02
• and erf is a special function. – Santosh Linkha Sep 24 '13 at 15:02
• @experimentX I have seen tables to evaluate this integrals, which imply that it can only be evaluated using numerical methods right?And what does you mean by special function? – dexterdev Sep 24 '13 at 15:08
• table is like a calculator to know it's value like you used log tables back in old days when there weren't any calculators. For special functions, see here. Erf or error function is just a type of special function in long list of it. – Santosh Linkha Sep 24 '13 at 15:22

$$\int_{a}^{b}e^{-x^2/2}dx= \sqrt {\frac{\pi}{2} }\left({{\rm erf}\left(\frac{b}{\sqrt{2}}\right)}-{{\rm erf}\left(\frac{a}{\sqrt{2}}\right)}\right),$$
where $\rm erf(x)$ is the error function
$$\operatorname{erf}(x) = \frac{2}{\sqrt{\pi}}\int_{0}^x e^{-t^2}\,\mathrm dt.$$
• @dexterdev: You will get a number once you specify $a$ and $b$. Also, note that, this is a special function not an elementary function. – Mhenni Benghorbal Sep 24 '13 at 15:22