The problem is Exercise 5.4.11 in Vershynin's book [High-Dimensional Probability](https://www.math.uci.edu/~rvershyn/papers/HDP-book/HDP-book.pdf). The book gives strong hints on how to solve it. It has also been discussed on [math.stackexchange.com](https://math.stackexchange.com/questions/3759461/an-elementary-application-of-integral-identity). Hope you can figure it out from here.