Help with this path integral. difficulty, something, function

amy

I am having troubles to solve the functional integral:
if a and b were 0 the integral is just a Gaussian integral but i do not know how to deal with the Delta distribution inside some may help ??  

    I have difficulty understanding what X is. Is it some variable, or is it a function, like X(x). When you write

\\int\\mathcal{D}X')">\\int\\mathcal{D}X '>

it looks like X is a function, for example  or  or something similar. But when you write

\\delta(X-3)')">\\delta(X-3) '>

it looks like X is just some parameter, like .

Or is the number 3 a constant function , , and the delta function an infinite dimensional delta function, like ?  

    if that is in path integral repreesetation, shouldn't there be an integral in the exponent.?, where S(q) is the action which is an integration over relevant time period of the Lagrangian of system.