# Is this identity containing the Gaussian Integral of any use?

• B
• MevsEinstein

#### MevsEinstein

TL;DR Summary
What the title says
I found this identity: ##x\int e^{-x^2} dx - \int \int e^{-x^2} dx dx = e^{-x^2}/2## by solving the integral of ##x*e^{-x^2}## and then finding its integration-by-parts equivalent. Is this identity useful at all?

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Summary:: What the title says

I found this identity: ##x\int e^{-x^2} dx - \int \int e^{-x^2} dx dx = e^{-x^2}/2## by solving the integral of ##x*e^{-x^2}## and then finding its integration-by-parts equivalent. Is this identity useful at all?
IMO, no. Unless I'm missing something, this looks similar to the integral ##\int_{-\infty}^\infty e^{-x^2}dx## is evaluated. IOW, by instead looking at the double integral. I'd bet this technique is in most calculus textbooks, although usually as the integral ##\int_{-\infty}^\infty \int_{-\infty}^\infty e^{(-x^2 - y^2)/2}dy dx##.

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• vanhees71
Fix notation. Using 'x' too much. For example dxdx?

• PhDeezNutz
IMO, no. Unless I'm missing something, this looks similar to the integral ##\int_{-\infty}^\infty e^{-x^2}dx## is evaluated. IOW, by instead looking at the double integral. I'd bet this technique is in most calculus textbooks, although usually as the integral ##\int_{-\infty}^\infty \int_{-\infty}^\infty e^{(-x^2 - y^2)/2}dy dx##.
There aren't any infinities in the formula.

Fix notation. Using 'x' too much. For example dxdx?
I don't know how to fix that problem. I'm still 13.

I don't know how to fix that problem. I'm still 13.
Use different letters for different things.

• I found this identity: ##x\int e^{-x^2} dx - \int \int e^{-x^2} dx dx = e^{-x^2}/2## by solving the integral of ##x*e^{-x^2}## and then finding its integration-by-parts equivalent. Is this identity useful at all?
$$\int xe^{-x^2}dx=-e^{-x^2}/2$$.
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