Is this identity containing the Gaussian Integral of any use?

[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?

 

[Mark44]

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 $\int_{-\infty}^\infty \int_{-\infty}^\infty e^{(-x^2 - y^2)/2}dy dx$.

 

[mathman]

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

 

[MevsEinstein]

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.

 

[MevsEinstein]

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

I don't know how to fix that problem. I'm still 13.

 

[mathman]

I don't know how to fix that problem. I'm still 13.

Use different letters for different things.

 

[malawi_glenn]

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?

Should read
$x\int e^{-x^2} dx - \int \int e^{-y^2} dydx = -e^{-x^2}/2$ looks like also a sign error.
These "identities" can be generalized to many functions, its just partial integration and change of variables, for instance consider xsin(x²)

 

[mathman]

$$\int xe^{-x^2}dx=-e^{-x^2}/2$$.

 

[MevsEinstein]

looks like also a sign error.

I couldn't edit the OP.