Projection kernel determinant integration ($\int D_{n}d\mu (x_n)=(r-n+1)D_{n-1}$)

Suppose an integral kernel reproduces under convolution and has integrable diagonal with integral $r$, and the displayed single-variable integrals exist. In the determinant $D_n=\det [K(x_i,x_j)]$, permutations fixing $n$ contribute $r{D}_{n-1}$ after integration. For every permutation on $n-1$ letters there are $n-1$ ways to insert $n$ into a cycle; convolution contracts the two adjacent factors, and insertion reverses the permutation sign. These terms contribute $-(n-1)D_{n-1}$. This proves the formula without a symmetry assumption on the kernel.