Integral of product of $h_{x_{i}}(x)$ and $h_{x_{j}}(x)$, where $h_{c}(x) = \min(c,x)'$

Question

I am trying to integrate the product of $h_{x_{i}}(x)$ and $h_{x_{j}}(x)$, where $ h_{c}(x) = \min(c,x)'$ as follows:

$$ h_{c}(x) = [\min(c,x)]' $$

$$ \int_0^\infty h_{x_{i}}(x)h_{x_{j}}(x)\,\mathrm dx = \min(x_{i},x_{j}) $$

Any help would be great!