Integrating a Gaussian Pulse for Kinetic Energy Calculation

[atqamar]

I'm having a hard time evaluating this integral.

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

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

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 $\int x^2 e^{-x^2}dx$ notice that this is the same as $\int x (x e^{-x^2})dx$ where the term between brackets is easy to integrate. Now use partial integration.

 

[atqamar]

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