Solution
= Solution
For $x,y>0$, the <Laplace transform> identity
$$
\frac1{x+y}=\int_0^\infty e^{-tx}e^{-ty}\,dt
$$
exhibits $k_1$ as an inner product of the <feature functions> $t\mapsto e^{-tx}$ in $L^2(0,\infty)$. More explicitly, for any real $c_i$ and positive $x_i$,
$$
\sum_{i,j}c_ic_jk_1(x_i,x_j)
=\int_0^\infty\left(\sum_i c_i e^{-tx_i}\right)^2dt\geq0.
$$
Therefore $k_1$ is a <positive-semidefinite kernel>.