Is there only numerical method to find this integral or any other way? Any references or links would be helpful.
$$\int _{a} ^ {b} e ^ {-x^2/2} dx .$$
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$a$ and $b$ are constants – dexterdev Sep 24 '13 at 15:00
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You can get an answer in terms of error function. – Mhenni Benghorbal Sep 24 '13 at 15:02
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and erf is a special function. – S L Sep 24 '13 at 15:02
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@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
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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. – S L Sep 24 '13 at 15:22
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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
$$ \operatorname{erf}(x) = \frac{2}{\sqrt{\pi}}\int_{0}^x e^{-t^2}\,\mathrm dt. $$
Mhenni Benghorbal
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@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
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Benghoral : But how do we find error function. Is there any method to calculate this integral. – dexterdev Sep 24 '13 at 16:18
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