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The integral $\int_0^\infty e^{-t^2}dt$ [duplicate]

Me and my highschool teacher have argued about the limit for quite a long time. We have easily reached the conclusion that integral from $0$ to $x$ of $e^{-t^2}dt$ has a limit somewhere between $0$ ...
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How do I evaluate the following integral $\int_{-\infty}^{\infty} e^{-\sigma^2 x^2/2}\; \mathrm dx$? [duplicate]

How do I evaluate the following integral $$\int_{-\infty}^{\infty} \exp\left(-\frac{\sigma^2 x^2}{2}\right) \mathrm dx\;?$$ How is it even possible to find an antiderivative? The integral is ...
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Computing the integral of $e^{-x^2}$ over the entire line [duplicate]

Possible Duplicate: Proving $\\int_{0}^{+\\infty} e^{-x^2} dx = \\frac{\\sqrt \\pi}{2}$ At lunch with a math friend years ago, he showed me an integral whose solution was, he said, so beautiful ...
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On evaluating $\int_0^\infty e^{-x^2}\,dx$ [duplicate]

Are there any other methods to integrate the following integral than the ones cited on the Wikipedia page? Maybe with a more elementary tool set? $$\int_0^\infty e^{-x^2}\,dx$$ Addendum: I am not ...
Integral $\int_{0}^{\infty} e^{-4x^2}dx$ [duplicate]
I'm trying to solve this $$\int_{0}^{\infty} e^{-4x^2}dx$$ I saw how $\int_{\infty}^{\infty} e^{-x^2}dx$ it's done and how it covers the whole plane, then I assume that from $0 \ to \ \infty$ it's ...
Integrating $\int_{-\infty}^{\infty} e^{-2b x^2} \, dx$ [duplicate]
I need help in integrating this integral: $$\int_{-\infty}^{\infty} e^{-2b x^2} \, dx$$ Knowing the value of $\int_{-\infty}^{\infty} e^{-x^2} \, dx$, and the integral for $e^{bx}$, I'm guessing ...