# Manipulating a product term inside an integral

I have an expression of the form $$P(x) = \int_0^\infty \prod_{i=0}^{n} e^{-G(a, x_i)}\,\mathrm{d}a$$ and I was wondering if there was any way that I could swap the order of the product and the integral? I suspect not but its been a while since I had to manipulate integrals for myself. Ideally P(x) will represent the total probability of making i measurements from a Possion distributed system with zero occurences each time.

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In general, $\int fg \not= \int f \int g$. –  AlexE Jan 16 '12 at 10:51
Thanks, I thought that was probably the case. –  Bowler Jan 16 '12 at 11:43
1. I think you mean $\prod_{i=0}^N$?! 2. Would it help to write $e^{-\sum G }$? –  draks ... Jan 16 '12 at 13:29

This means that in general, evaluating integrals like the one stated can be quite hard, especially if the mapping $G$ is not in some way "nice".