# analytic expression for integral weighted average of exponential function

Based on some numerical calculations I've done it appears that the following is true:

$\frac{\int w(x) \exp(-f(x) z) dx}{\int w(x) dx} = a \exp(-b z) + c$

where $w(x)$ is a function with a single peak (such as a Gaussian or Lorentzian spectral lineshape) and f(x) can be quite broad in x. The left hand side can be interpreted as the weighted average of $\exp(-f(x) z)$ with weighting function $w(x)$. Is there a proof or analytical demonstration that the above expression is correct? If so, are there expressions for the constants $a, b, c$ on the right hand side in terms of f(x) and w(x)?

Edit: We can reduce the number of constants by one: when $z=0$ the left hand side is 1 and the right hand side is $a+c$. Therefore we can substitute $c=1-a$.