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

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

  • $\begingroup$ $a$ and $b$ are constants $\endgroup$ – dexterdev Sep 24 '13 at 15:00
  • $\begingroup$ You can get an answer in terms of error function. $\endgroup$ – Mhenni Benghorbal Sep 24 '13 at 15:02
  • $\begingroup$ and erf is a special function. $\endgroup$ – Santosh Linkha Sep 24 '13 at 15:02
  • $\begingroup$ @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? $\endgroup$ – dexterdev Sep 24 '13 at 15:08
  • $\begingroup$ 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. $\endgroup$ – Santosh Linkha Sep 24 '13 at 15:22

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

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

  • $\begingroup$ Sir, But I was looking for a number. $\endgroup$ – dexterdev Sep 24 '13 at 15:13
  • $\begingroup$ @dexterdev: You will get a number once you specify $a$ and $b$. Also, note that, this is a special function not an elementary function. $\endgroup$ – Mhenni Benghorbal Sep 24 '13 at 15:22
  • $\begingroup$ Benghoral : But how do we find error function. Is there any method to calculate this integral. $\endgroup$ – dexterdev Sep 24 '13 at 16:18
  • $\begingroup$ ok got it...Error function is evaluated using McLaurin series. $\endgroup$ – dexterdev Sep 25 '13 at 8:33

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