# Integral of x^{2} e^{-x^2} dx

by niks
Tags: integral
 P: 2 I ran into an integral while working on response of a signal processing filter, it looks like: $$\int_{-\infty}^{\infty} x^{2} e^{-x^{2}} dx$$ While trying integration by parts u = $$x^{2}$$ we get du = 2xdx but can't proceed with dv = $$e^{-x^{2}}$$ because then v = $$\int e^{-x^{2}}$$ can't be integrated unless we use the limits. Can anyone suggest an approach for this? Thanks, Niks
 Math Emeritus Sci Advisor Thanks PF Gold P: 38,886 What exactly are you trying to do? As you point out, your v is not any elementary function, and that tells you that neither is $$\int x^2e^{-x^2} dx$$ You might be able to do that in terms of the "error function", Erf(x), which is defined to be $$\int e^{-x^2} dx$$
 P: 64 If you just want to calculate the definite integral, I don't see why you wouldn't want to include the limits when integrating by parts?
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PF Gold
P: 38,886

## Integral of x^{2} e^{-x^2} dx

His point, about the limits of integration, was that it is well known that
$$\int_{-\infty}^{\infty} e^{x^2}dx= 2\sqrt{\pi}[/itex] while the anti-derivative is not an elementary function. in other words, he could, theoretically, do it as a definite integral with the right limits but not as an indefinite integral.  P: 64 Was that adressed to me? Anyway, Maple tells me that: [tex] \int_{-\infty}^{\infty} x^{n} e^{-x^{2}}\rm{d} x=\frac{1}{2}\Gamma\left(\frac{n}{2}+\frac{1}{2}\right)\left(1+(-1)^n\right)$$ , which should be possible to prove by induction. PS. HallsofIvy: You have forgotten the minus-sign in your integrand.
 Sci Advisor HW Helper P: 1,276 A cute way to solve this is to recall that $$\int_{-\infty}^{\infty}e^{-ax^2}=\frac{\sqrt{\pi}}{\sqrt{a}}$$ Then use Feynman's favorite trick and differentiate both sides with respect to a, and evaluate at a = 1.
P: 85
 Quote by HallsofIvy His point, about the limits of integration, was that it is well known that $$\int_{-\infty}^{\infty} e^{x^2}dx= 2\sqrt{\pi}[/itex] while the anti-derivative is not an elementary function. in other words, he could, theoretically, do it as a definite integral with the right limits but not as an indefinite integral. For one, it's $e^{-x^2}$ and for two, it's $\sqrt{\pi}$. P: 2  As you point out, your v is not any elementary function, I think I should use this as a guideline for future problems. Both u and v should be elementary functions otherwise integration by parts becomes too messy(perhaps impossible).  while the anti-derivative is not an elementary function. in other words, he could, theoretically, do it as a definite integral with the right limits but not as an indefinite integral. Yes, that was what I had in mind. That's why I got stuck there.  Anyway, Maple tells me that: [tex] \int_{-\infty}^{\infty} x^{n} e^{-x^{2}}\rm{d} x=\frac{1}{2}\Gamma\left(\frac{n}{2}+\frac{1}{2}\r ight)\left(1+(-1)^n\right)$$ which should be possible to prove by induction.
Thanks!! That will help me move forward.

Thanks to everyone who replied, I learnt a lot from this thread.

-Niks
 P: 9 Using lots of substitutions and integration by parts I get this: $$\int x^{2}e^{-{x^2}}dx=xe^{x^{2}}\left[1-\sum_{n=1}^{\infty}\frac{\prod_{k=2}^{n}\left(2k-3\right)}{2^{n}x^{2n}}\right]$$ I would go over the derivation but LaTex is killing me.
 P: 64 Quite frankly, I think using Leibniz' rule, as suggested by Nicksauce, would by far be the simplest method in this case.
P: 181
 Quite frankly, I think using Leibniz' rule, as suggested by Nicksauce, would by far be the simplest method in this case.
Quite true.

But, if doing by parts, then the proper selection of u an dv is

$$u=x$$

$$dv= x*e^{-x^2}dx$$

and then things won't be so messy - however, it will involve the definite integral $\int^{\infty}_{-\infty}{ e^{-{x^2}}dx}$ which we know equals $\sqrt{\pi}$.
 P: 1,705 umm mathematica gives me $$\frac{\sqrt{\pi}}{2}$$ and for the indefinite : $$\frac{1}{4} \sqrt{\pi } \text{erf}(x)-\frac{1}{2} e^{-x^2} x$$
 P: 270 Well, perhaps the very simplest approach is to recognize that the integral is $\sqrt{\pi}$ times the variance of a Gaussian random variable with mean 0 and standard deviation $\frac{1}{\sqrt{2}}$. That's certainly all I'd bother doing in the signal processing context the OP mentioned.

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