How to integrate the following?
$$
\int_\infty^\infty xe^{-\alpha^2 x^2}\,dx
$$

Question

How to integrate the following?
$$
\int_\infty^\infty xe^{-\alpha^2 x^2}\,dx
$$

thomas5267 · Answer

This is the expectation value of position of a quantum harmonic oscillator in the ground state.

@ganeshie8

thomas5267 · Answer

It is clear that this integral evaluates to 0 because it is of the same form of the expectation value of Gaussian distribution with 0 mean.  How to make it mathematically rigorous is the question.

thomas5267 · Answer

F*** F*** F*** I am an idiot.  Use u=x^2, find the indefinite integral, substitute u=x^2 back into the indefinite integral and evaluate!  I was frightened by the infinities!