Integrating a Gaussian Pulse for Kinetic Energy Calculation

[atqamar]

I'm having a hard time evaluating this integral.

A Gaussian pulse [tex]\psi (y,t) = Ae^{-( \frac{y-ct}{a} )^2}[/tex] is traveling in an infinite string of linear mass density [tex]\rho[/tex], under tension [tex]T[/tex].

I know the Kinetic Energy is the integral of the partial: [tex]\frac{\rho}{2} \int_{-\infty}^{\infty} (\frac{\partial \psi}{\partial t})^2 dy[/tex]. I evaluate the partial, and this simplifies to [tex]\frac{\rho}{2} \int_{-\infty}^{\infty} (\frac{2c(y-ct)}{a^2} \psi)^2 dy[/tex].

I don't know where to proceed from here. I tried u-substitution, and integration by parts, with no success. I think the error function is useful in this, but we haven't covered this in the physics course yet.

 

[Cyosis]

Integral is of the form [itex]\int x^2 e^{-x^2}dx[/itex] notice that this is the same as [itex] \int x (x e^{-x^2})dx[/itex] where the term between brackets is easy to integrate. Now use partial integration.

 

[atqamar]

Oh stupid me. That wasn't tough at all. Thanks!