Probability of two random uniform variables

Suppose X is uniform [e,f] and Y is uniform [g,h]. We want to find the pdf of the X+Y for the general uniform random variables.

Using the characteristic equation we multiply both together and we need to take the inverse Fourier transformation. I do not know how to simplify it. Is there a trick I am not seeing?


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That seems like an awfully complicated method to use to derive this distribution. Is there a particular reason you're using that approach?
Yes, in this case, the characteristic function approach will only take extra steps to prove the necessity of taking an integral. So you may as well do it from the beginning: the cdf of X+Y equals

F(z) = Prob(X+Y <= z) = Int_e^f f_X(x) * Prob(X+Y <= z | X = x) dx = Int_e^f 1/(f-e) * Prob(Y <= max(z-x,0) | X = x) dx = ...