Is this matrix positive semi-definite?
Yes, this is similar to a Hilbert matrix, and a clever trick does it: Note that the $(j,k)$ element is $a_{jk}=\int_0^1 x^{j+k-1}\,dx$, and so $$\sum_{j,k} a_{jk}\bar u_ju_k=\int_0^1\Bigl|\sum_j u_jx^j\Bigr|^2x^{-1}\,dx\ge0.$$