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Suppose I have a function $F$ defined on $SO(n)$ by $F(U) = f(u_{11}),$ where $f$ is some nice function (and $u_{11}$ is the top left matrix coefficient of $U$). Is there some reasonably civilized way to compute the integral of $F$ with respect to the Haar measure on $SO(n)?$

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For a Haar-uniform matrix in $SO(n),$ the first row is a uniform point n $S^n,$ and so any particular matrix coefficient is distributed as described in the answers to this question. Call the distribution $G(x),$ so the answer to the OP is that that the integral of $F$ is just the expectation of $f$ with respect to $G.$

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