Random phase (?) approximation

[Irid]

Hello,
I've come across equations where people use the approximation

$$\int_0^1 \exp(f(x))\, dx \approx \exp \left( \int_0^1 f(x)\, dx\right)$$

I can see that this is correct if f(x) is small, one just uses exp(x) = 1+x+...
However, it appears that this approximation has a broader validity that that... How is it called (Random phase approximation??) and where could I find more info about it?

 

[welcomeblack]

I've seen something similar used in mean field theory to estimate the partition function of some difficult to calculate system. I think the particular step that reminds me of your equation is called the Bogoliubov Inequality.