Let be $f(x)$ a function which can be expressed as a linear combination of other functions: $$f(x)=f_1(x)+f_2(x)+...+f_n(x)$$ would it be possible to apply Monte Carlo integration to each one of the factors and mix them somehow in order to get the right result of $f(x)$? If it's so, how?
Example x 1-dimension: Let's say you want to integrate $f(x)$ between on the interval $[x0,x1]$ using a uniform Monte Carlo estimator: $$f(x)=f_1(x)+f_2(x)$$ $$f_1(x)=ax^2\\f_2(x)=bx+c$$ $$a,b,c\ \ \text{constants}$$
How would you proceed?
A key thing underlying many MC methods is that expected values of sums are the sum of the expected values:
E(X + Y) = E(X) + E(Y)
WHETHER OR NOT X and Y are INDEPENDENT!
I personally find that kind of amazing. What this means is you can do pretty much anything (including re-using samples between sum components) are you have a unbiased estimate.
So for your example estimate f1 and f2 separately and then just add the estimates.