Question

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

Answer 1

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

@ganeshie8

Answer 2

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.

Answer 3

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!