I have an integral I need to evaluate that contains a random variable? How would I go about doing something like that. For instance lets say I have a random variable $C$ with PDF $f_C(x)=1/N \; 0\le x\le N$ (Uniform distribution) and I have an integral $$ \int\limits_a^bC\mathrm{d}x $$
Can I evaluate an integral like the above? What if the integral was: $$ \int\limits_a^bCf(x)\mathrm{d}x $$ Could I evaluate that integral? If yes, then how?