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Building on the nice answer of Guillaume: The integral

\int_[-1,1]ⁿ \prod_i<j |x_i² - x_j²| dx₁...dx_n

has the closed-form evaluation

4ⁿ / \prod_k≤n \binom{2k}{k}.

This basically follows from the evaluation of the Selberg beta integral S_n(1/2,1,1/2).

Combined with modding out by a typo, we now arrive at the following product formula for the volume of the unit ball of nxn matrices in the matrix norm:

n! \prod_k≤n π^k / ((k/2)! \binom{2k}{k}).

In particular, we have:

• 2/3 π² for n=2
• 8/45 π⁴ for n=3
• 4/1575 π⁸ for n=4