# Gausian Integrals... How do you do this one?

#### chutsu

Trying to solve this Gaussian integral, any help please?

$$\displaystyle \int^{\infty}_{0} x e^{-x^2} dx$$

Thanks
Chris

#### running-gag

Hi

What is the derivative of $$\displaystyle e^{-x^2}$$ ?

#### chutsu

I believe the intergration of

$$\displaystyle e^(-x^2)$$

with limits 0 and $$\displaystyle \infty$$

is

$$\displaystyle \int^{\infty}_{0} e^{-x^2} = \sqrt{\pi}$$

#### running-gag

I was not asking for the integration of $$\displaystyle e^{-x^2}$$ but its derivative (Wink)

#### chutsu

$$\displaystyle -2xe^{-x^2}$$ ??

Am i missing something here?

#### running-gag

$$\displaystyle -2xe^{-x^2}$$ ??

Am i missing something here?
No that is right

Now you can compute $$\displaystyle \int_{0}^{+\infty} x e^{-x^2}dx$$

#### chutsu

throw me another bone please? lol

#### running-gag

The derivative of $$\displaystyle e^{-x^2}$$ is $$\displaystyle -2xe^{-x^2}$$

Therefore an antiderivative of $$\displaystyle -2xe^{-x^2}$$ is $$\displaystyle e^{-x^2}$$

And an antiderivative of $$\displaystyle xe^{-x^2}$$ is $$\displaystyle -\frac12 e^{-x^2}$$

• chutsu