Conversion of proper to improper integrals and vice versa, is the following case possible?

Question

I recently stumbled across a particular integral which got me asking myself this: Given a function (Smooth enough that we can play around with, and no convergence issues when integrating) $f(x)$, is there a function $g(x,T)$ satisfying:

$$ \int_0^T f(x) \, dx = \int_0^\infty g(x,T) \, dx $$

How do we then compute $g$