The identity derived from the integral of ##x e^{-x^2}##, expressed as ##x\int e^{-x^2} dx - \int \int e^{-x^2} dx dx = e^{-x^2}/2##, has limited utility in its current form. The discussion highlights that this identity resembles the Gaussian integral ##\int_{-\infty}^\infty e^{-x^2}dx## and suggests that it may be more effectively represented using double integrals. Additionally, there are concerns regarding notation, particularly the excessive use of 'x' and potential sign errors in the expression.
PREREQUISITESStudents of calculus, mathematicians interested in integral identities, and educators seeking to clarify integration techniques and notation.
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##.MevsEinstein said: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?
There aren't any infinities in the formula.Mark44 said: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##.
I don't know how to fix that problem. I'm still 13.mathman said:Fix notation. Using 'x' too much. For example dxdx?
Use different letters for different things.MevsEinstein said:I don't know how to fix that problem. I'm still 13.
Should readMevsEinstein said: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?
I couldn't edit the OP.drmalawi said:looks like also a sign error.