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