User:EGM6321.f10.team6.cook/integration question

$$ \begin{align} g(x,y) = {} & \int_{-\infty}^{+\infty}{b(x)c(y)dx}\\ & c(y)\int_{-\infty}^{+\infty}{b(x)dx}\\ = & c(y)\int_0^{\infty}{b(x)dx}+f(x) \end{align} $$

How to get to

$$ g(x,y) = c(y)\left(\int_0^{\infty}{b(x)dx}+f(x)\right) $$

without assuming $$g$$ is linear?