I would like to be able to translate
Exp[a+b]
to
Exp[a] Exp[b]
but when I execute
Exp[a+b]/.Exp[m_+n_]-> Exp[m] Exp[n]
Mathematica seems to re-combine the product. The reason for wanting to split the exponential is that I have an integral of the form
Integrate[Exp[f[x]+g[y]] h[y],y]
that I would like to evaluate symbolically to yield
Exp[f[x]] Integrate[Exp[g[y]] h[y],y]
I have determined that Mathematica can recognize when a factor that does not depend on the variable of integration can be pulled outside an integration so that
Integrate[Exp[x] f[y],y]
is evaluated as
Exp[f[x]] Integrate[f[y],y]
The above is the gist of my question, but a concrete example of what I am trying to get is to convert this
Integrate[Exp[(I T(l(-lam+th)+n(mu-w)))] P[lam] P[th] P[-mu+th] P[lam-w],th,lam]
to this
Exp[ I n(mu-w)] Integrate[Exp[(I T(l(-lam+th)))]P[lam] P[th] P[-mu+th] P[lam-w],th,lam]
I copied and pasted these equations from a notebook and tried to edit the result to make it more readable, so if there is a mismatched parenthesis the problem is my editing, not a fundamental error in the equation
EDIT::
I tried using HoldForm which did maintain the product of exponentials but the integration did not move the exponential outside the integration
mykern = Exp[u + v] h[v] /. Exp[a_ + b_] -> HoldForm[Exp[a] Exp[b]]
Integrate[mykern, v]
results in
Integrate[h[v] (Exp[u] Exp[v]),v]
and releasing the hold just on that result just changes back to an exponential of the sum